GMAT 数 学 笔 记. By Yumeng Guo. 来 自 Manhattan 系 列 5 本 书 与 OG13

Transcription

1 GMAT 数 学 笔 记 By Yumeng Guo 来 自 Manhattan 系 列 5 本 书 与 OG13 1

2 IMPORTANT FORMULAS 7 OG 知 识 点 9 Arithmetic 9 Properties of Integers 9 Real Numbers 9 Powers and Roots of Numbers 9 Descriptive Statistics 9 Sets 9 Discrete Probability 10 Algebra 11 Absolute Value 11 Functions 11 Geometry 11 Lines 11 Polygons (Convex) 11 Triangles 11 Coordinate Geometry 11 Word Problem 11 Discount 11 Sets 12 GUIDE 1 - THE FRACTIONS, DECIMALS & PERCENTS GUIDE 13 Chapter 1. Digits & Decimals 13 Digits 13 Place Value 13 Rounding to the Nearest Place Value 13 Chapter 2. Fractions 13 Numerator and Denominator Rules 13 Problem Set 13 Chapter 3. Percents 14 Percent Change vs. Percent of Original 14 Interest Formulas 14 Chapter 4. Ratios 14 The Unknown Multiplier 14 Chapter 5. FDPs 15 Common FDP Equivalents 15 2

3 When to Use Which Form 15 Chapter 6. FDP Strategies 16 The DS Process 16 Benchmark Values 16 Chapter 7. Extra FDPs 16 Repeating Decimals 16 Terminating Decimals 16 Unknown Digits Problems 17 Formulas That Act on Decimals Error! Bookmark not defined. Problem Set 17 GUIDE 2 - THE ALGEBRA GUIDE 19 Chapter 1. PEMDAS 19 Chapter 2. Linear Equations 19 Expressions vs. Equations 19 Absolute Value Equations 19 Chapter 3. Exponents 19 All about the Base 19 Factoring Out a Common Term 19 Equations with Exponents 20 Same Base or Same Exponent 20 Problem Set 20 Chapter 4. Roots 20 A Square Root Has Only One Value 20 Roots and Fractional Exponents 21 Simplifying a Root 21 Memorize: Squares and Square Roots 21 Memorize: Cubes and Cube Roots 21 Chapter 5. Quadratic Equations 21 Problem Set 21 Chapter 6. Formulas 22 Sequence Formulas 22 Recursive Sequences 22 Problem Set 22 Chapter 7. Inequalities 22 Much like Equations, With One Big Exception 22 Combining Inequalities: Add 'Em Up! 22 Problem Set 23 3

4 Chapter 8. Algebra Strategies 23 Problem Set 23 Chapter 9. Extra Equations Strategies 24 Complex Absolute Value Equations 24 Quadratic Formula 24 Chapter 10. Extra Functions Strategies 24 Common Function Types 24 Uncommon Function Types 25 Chapter 11. Extra Inequalities Strategies 25 Optimization Problems 25 Problem 25 GUIDE 5 - THE NUMBER PROPERTIES GUIDE 27 Chapter 1. Divisibility & Primes 27 Rules of Divisibility by Certain Integers 27 Fewer Factors, More Multiples 27 Primes 27 Greatest Common Factor and Least Common Multiple 27 Remainders 28 Three Ways to Express Remainders 28 Problem Set 29 Chapter 2. Odds, Evens, Positives, & Negatives 29 Arithmetic Rules of Odds & Evens 29 The Sum of Two Primes 29 Representing Evens and Odds Algebraically 30 Multiplying & Dividing Signed Numbers 30 Problem Set 30 Chapter 3. Combinatorics 30 Arranging Groups 30 Arranging Groups with Repetition: The Anagram Grid 31 Problem Set 31 Chapter 4. Probability 32 The 1 - x Probability Trick 32 Problem Set 32 Chapter 5. Number Properties Strategies 32 Testing Odd & Even Cases 32 Problem Set 32 Chapter 6. Extra Divisibility & Primes 33 4

5 Primes 33 Divisibility and Addition/Subtraction 33 Advanced GCF and LCM Techniques 34 Other Applications of Primes & Divisibility 35 Advanced Remainders 36 Counting Total Factors 37 Problem Set 37 Chapter 7. Extra Combinatorics & Probability 39 Disguised Combinatorics 39 Arrangements with Constraints 39 Combinatorics and the Domino Effect 39 Problem Set 40 GUIDE 4 - THE GEOMETRY GUIDE 42 Chapter 1. Polygons 42 Quadrilaterals: An Overview 42 Chapter 2. Triangles & Diagonals 42 Common Right Triangles 42 Similar Triangles 42 Problem Set 42 Chapter 3. Circles & Cylinders 43 Inscribed vs. Central Angles 43 Inscribed Triangles 43 Problem Set 43 Chapter 4. Lines & Angles 44 Exterior Angles of a Triangle 44 Chapter 5. Coordinate Plane 44 Positive and Negative Quadrants 44 Chapter 6. Geometry Strategies 44 Principles of Geometry 44 Problem Set 45 Chapter 7. Extra Geometry 45 Maximum Area of Polygons 45 Function Graphs and Quadratics 45 Perpendicular Bisectors 45 The Intersection of Two Lines 45 Problem Set 46 5

6 GUIDE 3 THE WORD PROBLEM GUIDE 47 Chapter 1. Algebraic Translations 47 Problem Set 47 Chapter 2. Rates & Work 47 Average Rate: Don't Just Add and Divide 47 Problem Set 47 Chapter 3. Statistics 48 Weighted Averages 48 Median: The Middle Number 48 Standard Deviation 48 Problem Set 49 Chapter 4. Consecutive Integers 50 Evenly Spaced Sets ( 等 差 数 列 ) 50 Properties of Evenly Spaced Sets 51 The Sum of Consecutive Integers 51 Chapter 5. Overlapping Sets 51 The Double-Set Matrix 51 Overlapping Sets and Algebraic Representation 52 2 Sets, 3 Choices: Still Double-Set Matrix 52 3-Sets, 2 Choices: Venn Diagrams 52 Problem Set 53 Chapter 6. Word Problem Strategies 53 Problem Set 53 Chapter 7. Extra Problem Types 54 Scheduling 54 Problem Set 54 Chapter 8. Extra Consecutive Integers 55 Products of Consecutive Integers and Divisibility 55 Sums of Consecutive Integers and Divisibility 55 Consecutive Integers and Divisibility 56 Problem Set 56 6

7 DS 题 目 一 定 记 得 what 题 与 is 题 的 区 别 70% discount 是 3 折 Important Formulas 公 差 为 d 的 等 差 数 列 : ( ) ( ) ( ) 等 差 数 列 的 平 均 数 即 为 ( ) 公 比 为 q 的 等 比 数 列 : ( ) ( ) ( ) ( ) 边 长 为 a 的 等 边 三 角 形 面 积 海 伦 - 秦 九 韶 公 式 三 角 形 面 积 ( )( )( ) 圆 锥 体 积 直 角 三 角 形 : 垂 直 的 两 条 线 的 顶 点 ( ) 点 ( ) 到 的 距 离 ( )( ) ( )( ) 没 有 意 义 韦 达 定 理 7

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9 OG 知 识 点 Arithmetic Properties of Integers 5 divided by 7 has the quotient 0 and the remainder 5 注 意 余 数 是 5 而 不 是 7 Every integer greater than 1 either is prime or can be uniquely expressed as a product of prime factors. Real Numbers All real numbers correspond to points on the number line and all points on the number line correspond to real numbers. Powers and Roots of Numbers Every positive number n has two square roots, one positive and the other negative. Descriptive Statistics The mode of a list of numbers is the number that occurs most frequently in the list. A list of numbers may have more than one mode. Range: the greatest value in the numerical data minus the least value. Sets Set is a collection of numbers or other objects. The objects are called the elements of the set. If S is a set having a finite number of elements, then the number of elements is denoted by S. Subset: 子 集 ; Union: 并 集 ( ) ; Intersection: 交 集 ( ) Two sets that have no elements in common are said to be disjoint or mutually exclusive: If S and T are mutually exclusive: 9

10 Discrete Probability An event E occurs, denoted by ( ), ( ) E or F is the set of outcomes in E of F or both, that is, E and F is the set of outcomes in both E and F, that is, ( ) ( ) ( ) ( ) 例 如 扔 骰 子,E 是 结 果 为 奇 数 的 事 件, ;F 是 质 数 事 件, ; ( ) ( ) ( ) ( ) ( ) ( ) If the event E and F is impossible, then E and F are said to be mutually exclusive events, ( ) ( ) ( ) ( ) The two events A and B are said to be independent if the occurrence of wither event does not alter the probability that the other event occurs. Any independent events E and F: ( ) ( ) ( ) Q Events A and B are mutually exclusive and Events B and C are independent. ( ) ( ) ( ) S ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 即 和 不 ( ) ( ) ( ) ( ) Conclude: ( ) ( ) 10

11 Algebra Absolute Value Functions In any function there can be no more than one output for any given input. However, more than one input can give the same output. // 一 个 x 只 能 对 应 一 个 y, 但 一 个 y 可 对 应 多 个 x// Geometry Lines The notation is used to denote line segment PQ, and PQ is used to denote the length of the segment. Polygons (Convex) The sum of the interior angle measures of a polygon: ( ) Triangles An isosceles triangle has at least two sides of the same length. // isosceles triangle 也 可 能 是 等 边 三 角 形 // Coordinate Geometry Given any two points ( ) ( ) ( ) Word Problem Discount Q The price of an item is discounted by 20 percent and then this reduced price is discounted by an additional 30 percent. These two discounts are equal to an overall discount of what percent? A 44%, 而 不 是 56% 11

12 Sets In a certain production lot, 40 percent of the toys are red and the remaining toys are green. Half of the toys are small and half are large. If 10 percent of the toys are red and small, and 40 toys are green and large, how many of the toys are red and large? Answer: 60 Red Green Total Small 10% 50% Large 50% Total 40% 60% 100% 12

13 Guide 1 - The Fractions, Decimals & Percents Guide Chapter 1. Digits & Decimals Digits There are only ten digits in our number system: 0, 1, 2, 3, 4, 5, 6, 7, 8, is a number composed of three digits: 3, 5, and 6. Place Value Every digit in a number has a particular place value depending on its location within the number. For example, in the number 452, the digit 2 is in the ones (or units ) place, the digit 5 is in the tens place, and the digit 4 is in the hundreds place. hundred billions, ten billions, one billions, hundred millions, ten millions, one millions, hundred thousands, ten thousands, thousands, hundreds, tens, units or ones, tenths, hundredths, thousandths, ten thousandths Rounding to the Nearest Place Value rounded to the nearest tenth equals 3.7 // 注 意 tenth 的 位 置 // Chapter 2. Fractions Proper fractions are those that fall between 0 and 1. Improper fractions are those that are greater than 1. Numerator and Denominator Rules Positive fractions: Adding the same number to both the numerator and the denominator brings the fraction closer to 1, regardless of the fraction s value. Problem Set Q Multiply both the numerator and denominator of a positive, proper fraction by 3. 对 问 题 意 义 的 理 解 : 对 一 个 正 的 真 分 数, 其 分 子 和 分 母 都 乘 以 3 13

14 Chapter 3. Percents Percent Change vs. Percent of Original 10% increase = 110% of the original 130% of the original = 30% increase 10% greater than = 110% of the original 130% of the original = 30% greater than 45% decrease = 55% of the original 75% of the original = 25% decrease 45% less than = 55% of the original 75% of the original = 25% less than Interest Formulas ( ) P = principal, r = rate (in decimal form), n = number of times per year, and t = number of years. Q A bank account with $100 earns 8% annual interest, compounded quarterly. If there are no deposits or withdrawals, how much money will be in the account after 6 months? S ( ) Chapter 4. Ratios The two partners spend time working in the ratio of 1 to 3. For every 1 hour the first partner works, the second partner works 3 hours. The Unknown Multiplier Q A recipe calls for amounts of lemon juice, wine, and water in the ratio of 2:5:7. If all three combined yield 35 milliliters of liquid, how much wine was included? S 14

15 Chapter 5. FDPs Common FDP Equivalents When to Use Which Form Prefer fractions for doing multiplication or division. Prefer decimals and percents for doing addition or subtraction, for estimating numbers, or for comparing numbers. 15

16 Chapter 6. FDP Strategies The DS Process Step 1: Separate additional info from the actual question. Step 2: Determine whether the question is Value or Yes/No. Value: The question asks for the value of an unknown (e.g., What is x?). A statement is Sufficient when it provides 1 possible value. A statement is Not Sufficient when it provides more than 1 possible value. Yes/No: The question that is asked has two possible answers: Yes or No (e.g., Is x even?). A statement is Sufficient when it provides a definite Yes or definite No. A statement is Not Sufficient when the answer could be Yes or No. Step 3: Decide exactly what the question is asking. Step 4: Use the Grid to evaluate the statements. Benchmark Values Q Which is greater? S 第 一 个 小 于 0.5, 第 二 个 大 于 0.5 Chapter 7. Extra FDPs Repeating Decimals If the denominator is 9, 99, 999 or another number equal to a power of 10 minus 1, then the numerator gives you the repeating digits (perhaps with leading zeroes). Again, you can always find the decimal pattern by simple long division. Terminating Decimals If, after being fully reduced, the denominator has any prime factors besides 2 or 5, then its decimal will not terminate. If the denominator only has factors of 2 and/or 5, then the decimal will terminate. Terminate: 1/256, 256=2^8; 231/660=7/20, 20=2*2*5 Not terminate: 100/27, 27=3^3; 7/105=1/15, 15=3*5 16

17 Unknown Digits Problems Q S Problem Set Q A professional gambler has won 40% of his 25 poker games for the week so far. If, all of a sudden, his luck changes and he begins winning 80% of the time, how many more games must he play to end up winning 60% of all his games for the week? A 25 S 注 意 第 一 句 话 的 理 解, 本 意 为 开 始 玩 了 25 次, 赢 了 40%, 后 面 继 续 玩 第 26, 27 次 而 不 是 本 周 总 共 要 玩 25 次 17

18 Q What is the length of the sequence of different digits in the decimal equivalent of 3/7? A 6 S 注 意 题 目 的 理 解 Q A grocery store sells two varieties of jellybean jars, and each type of jellybean jar contains only red and yellow jellybeans. If Jar B contains 20% more red jellybeans than Jar A, but 10% fewer yellow jellybeans, and Jar A contains twice as many red jellybeans as yellow jellybeans, by what percent is the number of jellybeans in Jar B larger than the number of jellybeans in Jar A? A 10% S 注 意 是 B 比 A 多 了 A 的 百 分 之 多 少 A B red x 1.2x yellow y 0.9y X=2y 得 B: 3.3y A: 3y 18

19 Guide 2 - The Algebra Guide Chapter 1. PEMDAS Parentheses-Exponents-(Multiplication/Division)-(Addition/Subtraction) Chapter 2. Linear Equations Linear equations are equations in which all variables have an exponent of 1. Expressions vs. Equations The most basic difference between expressions and equations is that equations contain an equals sign, and expressions do not. Absolute Value Equations n + 9-3n = 3, n+9=3n+3 or -3n-3, n=3 or -3 // 记 得 返 回 验 证,-3 不 可 以 // Chapter 3. Exponents The mathematical expression 4 3 consists of a base (4) and an exponent (3). The expression is read as four to the third power All about the Base 0 raised to any power equals 0 Anything Raised to the Zero Equals One (0 0 is undefined) : Take the square root of the first number and then raise that value to the power of the second number. // 注 意 其 表 述 方 法 // Factoring Out a Common Term Q If x = , what is the largest prime factor of x? A 7 19

20 S ( ) Equations with Exponents Any even exponents in an equation make it dangerous, as it is likely to have 2 solutions. Same Base or Same Exponent Q Solve the following equation for w: (4 w ) 3 = 32 w-1 A -5 S 两 边 都 改 写 为 以 2 为 base If 0 x = 0 y or 1 x =1 y you cannot claim that x=y Problem Set Q If 4 a + 4 a+1 = 4 a+2-176, what is the value of a? A a=2 S 提 取 4 a Q If m and n are positive integers and (2 18 ) (5 m ) = (20 n ), what is the value of m? S 2 18 *5 m = (2*2*5) n =2 2n *5 n, 得 18=2n, m=n Chapter 4. Roots, x <2, -2<x<2 A Square Root Has Only One Value Compare the following two equations: 对 比 There are two solutions to the equation on the left: x = 4 or x = -4. There is one solution to the equation on the right: x = 4. 20

21 If an equation contains a square root, only use the positive root. Odd roots (cube root, 5th root, 7th root, etc.) also have only one solution. Roots and Fractional Exponents Within the exponent fraction, the numerator tells you what power to raise the base to, and the denominator tells you which root to take. You can raise the base to the power and take the root in either order. ( ) ( ) ( ) Simplifying a Root Memorize: Squares and Square Roots Memorize: Cubes and Cube Roots Chapter 5. Quadratic Equations Problem Set Q What is x? (1) 21

22 (2) A E (x+8)(x-4)=0 S X 得 到 两 个 确 切 值 只 有 得 到 一 个 确 切 值 才 能 推 出 题 干 Chapter 6. Formulas Sequence Formulas Recursive Sequences If, what is the value of a 4 Problem Set Q The "competitive edge" of a baseball team is defined by the formula where W represents the number of the team's wins, and L represents the number of the team's losses. This year, the GMAT All-Stars had 3 times as many wins and onehalf as many losses as they had last year. By what factor did their "competitive edge" increase? A The competitive edge has increased by a factor of. S 对 factor 的 理 解 Chapter 7. Inequalities Much like Equations, With One Big Exception Multiply or divide an inequality by a negative number, the inequality sign flips. You cannot multiply or divide an inequality by a variable, unless you know the sign of the number that the variable stands for. Combining Inequalities: Add 'Em Up! 不 等 式 的 加 减 乘 除 Adding inequalities together is a powerful technique on the GMAT. However, note that you should never subtract or divide two inequalities. Moreover, you can only multiply inequalities together under certain circumstances. Only multiply inequalities together if both sides of both inequalities are positive. 22

23 // 例 如,m<2, n<5, 只 有 在 它 俩 都 大 于 0 的 情 况 下, 才 能 得 出 mn<10// Problem Set Q If, which of the following must be true? A A ( ) ( ) ( ) ( ) ( ) Q If is negative? A D (1) (2) Chapter 8. Algebra Strategies Problem Set Q If, what is the value of positive integer p? A B (1) (2) Q A retailer sells only radios and clocks. If she currently has 44 total items in inventory, how many of them are radios? A C (1) The retailer has more than 28 radios in inventory (2) The retailer has less than twice as many radios as clocks in inventory. S 注 意 (2) 的 意 义 是 r<2c 23

24 Chapter 9. Extra Equations Strategies Complex Absolute Value Equations Q If x-2 = 2x-3, what are the possible values for x? S Case A: Same sign: x-2=2x-3 Case B: Different sign: x-2=-(2x-3) Quadratic Formula For any quadratic equation of the form The solutions for x: Discriminant ( ) can convey important information: Chapter 10. Extra Functions Strategies Common Function Types Direct proportionality relationships are of the form, where x is the input value and y is the output value, k is called the proportionality constant. Inverse proportionality relationships are of the form value and y is the output value, k is called the proportionality constant. Linear growth, where x is the input Q Jake was 4.5 feet tall on his 12th birthday, when he began to have a growth spurt. Between his 12th and 15th birthdays, he grew at a constant rate. If Jake was 20% taller on his 15th birthday than on his 13th birthday, how many inches per year did Jake grow during his growth spurt? (12 inches = 1 foot) 24

25 S 得 到 12th 13th 14th 15th x 4.5+2x 4.5+3x 4.5+3x=1.2(4.5+x) Uncommon Function Types Exponential Growth ( ) Q A quantity increases in a manner such that the ratio of its values in any two consecutive years is constant. If the quantity doubles every 6 years, by what factor does it increase in two years? S Chapter 11. Extra Inequalities Strategies Optimization Problems Q If, what is the maximum possible for? A 49 S Problem Q Is A E (1) (2) a b ab Min -7 Min Min -7 Max 8-56 Max 6 Min Max 6 Max 8 48 Q If what is the possible range of values for x? 25

26 A x<0, or x>12 Q If what is the possible range of values for x? A -12< x<0 Q Is (1) (2) A C 26

27 Guide 5 - The Number Properties Guide Chapter 1. Divisibility & Primes Rules of Divisibility by Certain Integers An integer is divisible by: 4 if the integer is divisible by 2 TWICE, or if the LAST TWO digits are divisible by 4. For example, 23,456 is divisible by 4 because 56 is divisible by 4 5 if the integer ends in 0 or 5. 6 if the integer is divisible by BOTH 2 and 3. 8 if the integer is divisible by 2 THREE TIMES, or if the LAST THREE digits are divisible by 8. 9 if the SUM of the integer s DIGITS is divisible by 9. The GMAT can also test these divisibility rules in reverse. For example, if you are told that a number has a ones digit equal to 0, you can infer that that number is divisible by 10. Similarly, if you are told that the sum of the digits of x is equal to 21, you can infer that x is divisible by 3 but not by 9. Fewer Factors, More Multiples 12 is divisible by 3 3 is a divisor of 12, or 3 is a factor of is a multiple of 3 3 divides 12 12/3 is an integer 12/3 yields a remainder of 0 12 = 3n, where n is an integer 3 goes into 12 evenly Primes The first ten prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. *OG-PQ-26: 大 于 3 的 质 数 : 6q+1 或 6q+5 Greatest Common Factor and Least Common Multiple Greatest Common Factor (GCF): the largest divisor of two or more integers. Least Common Multiple (LCM): the smallest multiple of two or more integers. 27

28 短 除 法 : 在 用 短 除 计 算 多 个 数 时, 对 其 中 任 意 两 个 数 存 在 的 因 数 都 要 算 出, 其 它 没 有 这 个 因 数 的 数 则 原 样 落 下 直 到 剩 下 每 两 个 都 是 互 质 关 系 Remainders Dividend = Quotient x Divisor + Remainder Three Ways to Express Remainders Q When positive integer A is divided by positive integer B, the result is Which of the following could be the remainder when A is divided by B? (A) 13 (B) 14 (C) 15 (D) 16 (E) 17 28

29 A B S 只 有 当 时, 才 为 整 数 // // Problem Set Q If j is divisible by 12 and 10, is j divisible by 24? S 看 12 和 10 的 最 小 公 倍 数 能 否 被 24 整 除 Manhattan 另 有 方 法 Chapter 2. Odds, Evens, Positives, & Negatives Arithmetic Rules of Odds & Evens When you multiply integers, if ANY of the integers is even, the result is EVEN. Likewise, if NONE of the integers is even, then the result is ODD. The Sum of Two Primes If you see a sum of two primes that is odd, one of those primes must be the number 2. Conversely, if you know that 2 cannot be one of the primes in the sum, then the sum of the two primes must be even. Q If x > 1, what is the value of integer x? A A (1) There are x unique factors of x. (2) The sum of x and any prime number larger than x is odd. 29

30 S 大 于 2 的 整 数 n 都 不 可 被 n-1 整 除,(1) 可 ;(2) 只 能 推 出 x 是 偶 数 题 目 中 没 说 x 是 质 数 Representing Evens and Odds Algebraically Q What is the remainder when a is divided by 4? (1) a is the square of an odd integer. (2) a is a multiple of 3. A A S ( ), 除 以 4, 余 数 是 1. Multiplying & Dividing Signed Numbers Q Is the product of all of the elements in Set S negative? (1) All of the elements in Set S are negative. (2) There are 5 negative numbers in Set S. A C S 只 有 (2) 不 行, 因 为 可 能 有 0 Problem Set Q If x, y, and z are prime numbers and x < y < z, what is the value of x? (1) xy is even. (2) xz is even A D Chapter 3. Combinatorics ( ) ( ) Arranging Groups // 即 从 开 始 乘 以 个 ; // The number of ways of arranging n distinct objects, if there are no restrictions, is n! (n factorial). 0!=1 30

31 Arranging Groups with Repetition: The Anagram Grid If m members of a group are identical, divide the total number of arrangements by m! Manhattan 算 组 合 的 方 式, 都 是 转 化 为 字 母 问 题, 分 子 为 字 母 个 数 的 阶 乘 ; 分 母 是 几 个 阶 乘 的 乘 积, 有 多 少 重 复 的 字 母, 就 有 多 少 个 乘 积 EEL: PGSSBBB: AAABBB: Q 7 people enter a race. There are 4 types of medals given as prizes for completing the race. The winner gets a platinum medal, the runner-up gets a gold medal, the next two racers each get a silver medal, and the last 3 racers all get bronze medals. What is the number of different ways the medals can be awarded? S P G S S B B B M way: My way: Problem Set Q In how many different ways can the letters in the word "LEVEL" be arranged? S Q Mario's Pizza has two choices of crust: deep dish and thin-and-crispy. The restaurant also has a choice of 5 toppings: tomatoes, sausage, peppers, onions, and pepperoni. Finally, Mario's offers every pizza in extra cheese as well as regular. If Linda's volleyball team decides to order a pizza with four toppings, how many different choices do the teammates have at Mario's Pizza? S 31

32 Chapter 4. Probability The 1 - x Probability Trick Q What is the probability that, on three rolls of a number cube with faces numbered 1 to 6, at least one of the rolls will be a 6? S ( ) Problem Set Q A magician has five animals in his magic hat: 3 doves and 2 rabbits. If he pulls two animals out of the hat at random, what is the chance that he will have a matched pair? A S 注 意 matched pair 是 相 同 的 动 物 Chapter 5. Number Properties Strategies Testing Odd & Even Cases Q If a, b, and c are integers and ab + c is odd, which of the following must be true? I. a + c is odd II. b + c is odd III. abc is even (A) I only (B) II only (C) III only (D) I and III (E) II and III A C A 偶 B 偶 C 奇 ab 偶, c 奇 A 偶 B 奇 C 奇 A 奇 B 偶 C 奇 ab 奇, c 偶 A 奇 B 奇 C 偶 Problem Set Q If x, y, and z are integers, is x even? A A (1) ( )( ) (2) 32

33 S Q If p, q, and r are integers, is pq + r even? (1) p + r is even (2) q + r is odd A E Q If c and d are integers, is c - 3d even? (1) c and d are odd. (2) c - 2d is odd. A A Q Is pqr > 0? (1) pq > 0 (2) q/r < 0 A E Chapter 6. Extra Divisibility & Primes Primes Q If x is a prime number, what is the value of x? A A (1) There are a total of 50 prime numbers between 2 and x, inclusive. (2) There is no integer n such that x is divisible by n and 1 < n < x. Divisibility and Addition/Subtraction If you add a multiple of N to a non-multiple of N, the result is a non-multiple of N. (The same holds true for subtraction.) 33

34 18-10 = 8 (Multiple of 3) - (Non-multiple of 3) = (Non-multiple of 3) *OG-PQ-77: If then n is divisible by which of the following? I. 15 II. 17 III.19 答 案 : 只 有 II If you add two non-multiples of N, the result could be either a multiple of N or a non-multiple of N = 32 (Non-multiple of 3) + (Non-multiple of 3) = (Non-multiple of 3) = 33 (Non-multiple of 3) + (Non-multiple of 3) = (Multiple of 3) The exception to this rule is when N= 2. Two odds always sum to an even. Q Is N divisible by 7? A C (1) n = x - y, where x and y are integers (2) x is divisible by 7, and y is not divisible by 7. S 注 意 回 答 的 是 能 否 给 题 干 肯 定 或 否 定 的 答 复 Advanced GCF and LCM Techniques (GCF of m and n) x (LCM of m and n) = mx n The GCF of m and n cannot be larger than the difference between m and n. Consecutive multiples of n have a GCF of n. ( 如 n=4: 8,12,16 是 ;8,16 不 是 ) o GCF of any two consecutive integers is 1 另 一 种 求 最 大 公 约 数 和 最 小 公 倍 数 的 方 法 ( 例 如 三 个 数 为 100,140,250): To calculate the GCF, take the smallest count (the lowest power) in any column. To calculate the LCM, take the largest count (the highest power) in any column. 34

35 Q Is the integer z divisible by 6? A A S For (1) (1) The greatest common factor of z and 12 is 3. (2) The greatest common factor of z and 15 is z - 3 m 即 便 z 有 别 的 因 数, 也 不 能 算 到 GCF 中, 因 为 12 没 有 因 此 z 必 然 不 可 被 6 整 除 注 意 :z 必 然 或 必 然 不 能 被 6 整 除, 都 可 以 推 出 题 干 For (2) z 3 n 5 m 2 t Q If the LCM of a and 12 is 36, what are the possible values of a? A 可 能 为 9, 18, a 2 0,2 1, Other Applications of Primes & Divisibility How many different prime factors? 3 (2, 5, 7) How many total prime factors (length)? 3+2+1=6 How many total factors? Includes all factors, not necessarily just prime factors (include 1) All perfect squares have an odd number of total factors. Similarly, any integer that has an odd number of total factors must be a perfect square. 35

36 The prime factorization of a perfect square contains only even powers of primes. It is also true that any number whose prime factorization contains only even powers of primes must be a perfect square. // 一 个 perfect squares 有 奇 数 个 因 数, 反 之 亦 然 ;perfect squares 的 prime factorization 中 指 数 都 是 偶 数, 反 之 亦 然 // 类 似, 判 断 是 否 是 perfect cube, 看 其 prime factorization 的 指 数 是 否 都 是 3 的 倍 数 Q If k 3 is divisible by 240, what is the least possible value of integer k? (A) 12 (B) 30 (C) 60 (D) 90 (E) 120 S 因 此 的 因 数 至 少 也 含 有 这 些 而 的 质 因 数 的 指 数 必 为 的 倍 数 因 此 最 小 为 Advanced Remainders When you divide an integer by 7, the remainder could be 0, 1, 2, 3, 4, 5, or 6. When you divide an integer by a positive integer N, the possible remainders range from 0 to (N 1). There are thus N possible remainders. Q If a / b yields a remainder of 5, c / d yields a remainder of 8, and a, b, c and d are all integers, what is the smallest possible value for b + d? A 6+9=15 If x leaves a remainder of 4 after division by 7, and y leaves a remainder of 2 after division by 7, x+ y leaves a remainder of = 6 after division by 7. If x leaves a remainder of 4 after division by 7, and z leaves a remainder of 5 after division by 7, x+ z leaves a remainder of = 2 after division by 7. x z leaves a remainder of = 6 after division by 7. 36

37 x * z leaves a remainder of 4 * 5 = 20 7 * 2 = 6 after division by 7. Counting Total Factors Example, How many different factors does 2000 have? 在 2000 的 因 数 中,2 的 指 数 可 能 是 0,1,2,3,4;5 的 指 数 可 能 是 0,1,2,3 因 此 2000 有 20 个 If a number has prime factorization (where a, b, and c are all prime), then the number has ( )( )( ) different factors. Problem Set Q If y = 30p, and p is prime, what is the greatest common factor of y and 14p, in terms of p? A 2p S 或 用 短 除 法 p p p p p Q Is p divisible by 168? (1) p is divisible by 14 (2) p is divisible by 12 A E 12 和 14 最 小 公 倍 数 84 Q Is pq divisible by 168? (1) p is divisible by 14 (2) q is divisible by 12 A C 37

38 Q What is the greatest common factor of x and y? (1) x and y are both divisible by 4. (2) x - y = 4 A C Q What is the value of integer x? (1) The least common multiple of x and 45 is 225. (2) The least common multiple of x and 20 is 300. A C S (1) 推 出 x 不 可 能 含 x 3 0, 3 1, or x 2 0, 2 1, or Q If x 2 is divisible by 216, what is the smallest possible value for positive integer x? A 36 S 因 此 最 小 Q If x and y are positive integers and x / y has a remainder of 5, what is the smallest possible value of xy? A 30 S x=5, y=6 38

39 Chapter 7. Extra Combinatorics & Probability Disguised Combinatorics 要 把 一 些 问 题 转 化 为 排 列 组 合 的 算 法 Q Alicia lives in a town whose streets are on a grid system, with all streets running east-west or north-south without breaks. Her school, located on a corner, lies three blocks south and three blocks east of her home, also located on a corner. If Alicia only walks south or east on her way to school, how many possible routes can she take to school? S 三 次 东 三 次 南 的 组 合 :, 即 EEESSS, 把 3 个 E 排 在 6 个 座 位 上 M way: EEESSS 的 组 合 问 题 : Arrangements with Constraints Q Greg, Marcia, Peter, Jan, Bobby, and Cindy go to a movie and sit next to each other in 6 adjacent seats in the front row of the theater. If Marcia and Jan will not sit next to each other, in how many different arrangements can the six people sit? S 简 化 一 下 问 题, 即 ABCDEF 六 个 人 排 位,AB 不 能 挨 着 将 AB 视 为 整 体,AB 挨 着 的 情 况 有, 而 且 AB 可 以 换 一 下 位 置, 故 AB 挨 着 有 240 可 能 性 AB 不 能 挨 着 的 情 况 为 Combinatorics and the Domino Effect Q A miniature gumball machine contains 7 blue, 5 green, and 4 red gumballs, which are identical except for their colors. If the machine dispenses three gumballs at random, what is the probability that it dispenses one gumball of each color? S 39

40 My way: M way: blue first, then green, then red:. 总 共 有 6 中 这 样 的 情 况, 再 乘 以 6 即 可 这 两 种 方 法 在 思 路 上 是 本 质 不 同 的, 前 者 是 满 足 条 件 的 情 况 数 量 / 总 数 量, 用 排 列 组 合 的 方 法 来 解 决 概 率 问 题 ; 后 者 是 多 米 诺 法 Problem Set Q Three gnomes and three elves sit down in a row of six chairs. If no gnome will sit next to another gnome and no elf will sit next to another elf, in how many different ways can the elves and gnomes sit? S GEGEGE or EGEGEG: (3*2*3*2)*2=72 Q Gordon buys 5 dolls for his 5 nieces. The gifts include two identical Sun-and-Fun beach dolls, one Elegant Eddie dress-up doll, one G.l. Josie army doll, and one Tulip Troll doll. If the youngest niece does not want the G.l. Josie doll, in how many different ways can he give the gifts? S My way: S,S,E,G,T 给 5 个 人 (1) 假 设 最 小 的 人 拿 S, (2) 最 小 的 人 不 拿 S: 因 此 总 共 48 种 M way: 首 先 不 考 虑 限 定 条 件,SSEGT 总 共 有 种 最 小 的 人 拿 S, 则 种, 一 减 即 可 Q Five A-list actresses are vying for the three leading roles in the new film, "Catfight in Denmark. The actresses are Julia Robards, Meryl Strep, Sally Fieldstone, Lauren Bake-all, and Hallie Strawberry. Assuming that no actress has any advantage in getting any role, what is the probability that Julia and Hallie will star in the film together? S 40

41 Q For one roll of a certain die, the probability of rolling a two is 1/6. If this die is rolled4 times, which of the following is the probability that the outcome will be a two at least 3 times? S ( ) ( ) 41

42 Guide 4 - The Geometry Guide Chapter 1. Polygons Quadrilaterals: An Overview Square is a special type of parallelogram that is both a rectangle and a rhombus. 3 Dimensions: Volume Q How many books, each with a volume of 100 in3, can be packed into a crate with a volume of 5,000 in3? S 不 知 道, 因 为 每 本 书 的 尺 寸 不 确 定 Chapter 2. Triangles & Diagonals 边 长 为 a 的 等 边 三 角 形 面 积 : Common Right Triangles 3-4-5, , , , Similar Triangles If two similar triangles have corresponding side lengths in ratio a: b, then their areas will be in ratio a 2 : b 2. Problem Set 42

43 Q In Triangle ABC, AD = DB = DC (see figure). Given that angle DCB is 60 and angle ACD is 20, what is the measure of angle x? A 10 Chapter 3. Circles & Cylinders Inscribed vs. Central Angles An inscribed angle is equal to half of the arc it intercepts. Inscribed Triangles If one of the sides of an inscribed triangle is a diameter of the circle, then the triangle must be a right triangle. Problem Set Q A circular lawn with a radius of 5 meters is surrounded by a circular walkway that is 4 meters wide (see figure). What is the area of the walkway? A 56pi 43

44 Chapter 4. Lines & Angles Exterior Angles of a Triangle An exterior angle of a triangle is equal in measure to the sum of the two nonadjacent (opposite) interior angles of the triangle. Chapter 5. Coordinate Plane Positive and Negative Quadrants Chapter 6. Geometry Strategies Principles of Geometry Don t assume what you don t know for sure. Q In the figure above, what is the value of x+y? S Lines m and n look parallel. But you have not been given any indication that they are. Without additional information, you are unable to answer the question. 44

45 Problem Set Q If O represents the center of a circular clock and the point of the clock hand is on the circumference of the circle, does the shaded sector of the clock represent more than 10 minutes? (1) The clock hand has a length of 10. (2) The area of the sector is more than 16pi. S 答 案 为 E 注 意 是 Yes/No 的 问 题, 如 果 确 定 小 于 10 度, 也 可 以 推 出 题 干 的 问 题 Chapter 7. Extra Geometry Maximum Area of Polygons Both of these principles can be generalized for n sides Of all quadrilaterals with a given perimeter, the square has the largest area. Of all quadrilaterals with a given area, the square has the minimum perimeter. If you are given two sides of a triangle or parallelogram, you can maximize the area by placing those two sides perpendicular to each other. Function Graphs and Quadratics ( ), 通 过 判 断 是 否 大 于 来 判 断 和 轴 交 点 个 数 Perpendicular Bisectors Perpendicular lines have negative reciprocal slopes. The Intersection of Two Lines Two linear equations can represent two lines that intersect at a single point, or they can represent parallel lines that never intersect. There is one other possibility: the two equations might represent the same line. In this case, infinitely many points (.x, y) along the line satisfy the two equations. 45

46 Problem Set Q The line represented by the equation is the perpendicular bisector of the line segment AB. If A has the coordinates( ), what are the coordinates for B? S AB 的 斜 率 为 0.5, 得 到 AB 的 方 程, 两 条 线 交 点 ( ) 及 AB 的 中 点, 由 此 得 到 B 点 为 ( ) Q What are the coordinates for the point on Line AB that is three times as far from A as from B, and that is in between points A and B? S 两 个 小 三 角 形 相 似, 三 个 对 应 边 比 例 相 等, 得 到 该 点 为 ( ) 46

47 Guide 3 The Word Problem Guide Chapter 1. Algebraic Translations Problem Set Q A circus earned $150,000 in ticket revenue by selling 1,800 V.I.P. and Standard tickets. They sold 25% more Standard tickets than V.I.P. tickets. If the revenue from Standard tickets represents one-third of the total ticket revenue, what is the price of a V.I.P. ticket? S 1800 是 VIP 和 普 通 票 合 起 来, 而 不 是 VIP1800 张 答 案 125 Chapter 2. Rates & Work Average Rate: Don't Just Add and Divide If an object moves the same distance twice, but at different rates, the average rate will be closer to the slower of the two rates than to the faster. Problem Set Q Two hoses are pouring water into an empty pool. Hose 1 alone would fill up the pool in 6 hours. Hose 2 alone would fill up the pool in 4 hours. How long would it take for both hoses to fill up two-thirds of the pool? S 注 意 问 题 答 案 是 1.6h Q Twelve identical machines, running continuously at the same constant rate, take 8 days to complete a shipment. How many additional machines, each running at the same constant rate, would be needed to reduce the time required to complete a shipment by 2 days? S 注 意 看 题, 是 twelve 而 不 是 two, 答 案 为 4 Q Mary and Nancy can each perform a certain task in m and n hours, respectively. Is m<n? 47

48 (1) Twice the time it would take both Mary and Nancy to perform the task together, each working at their respective constant rates, is greater than m. (2) Twice the time it would take both Mary and Nancy to perform the task together, each working at their respective constant rates, is less than n. A D S 对 于 (1), 两 人 速 度 分 别 是 和, 速 度 之 和 为, 一 起 做 耗 时, Manhattan 方 法 : 如 果 两 人 都 以 Mary 的 速 度 来 做, 则 需 要 时 间, 而 现 在 要 多 于, 说 明 Nancy 慢,n>m 注 意, 若 能 得 出 m>n, 也 是 sufficient Chapter 3. Statistics Weighted Averages A weighted average of only two values will fall closer to whichever value is weighted more heavily. For instance, if a drink is made by mixing 2 shots of a liquor containing 15% alcohol with 3 shots of a liquor containing 20% alcohol, then the alcohol content of the mixed drink will be closer to 20% than to 15%. Median: The Middle Number For sets containing an odd number of values, the median is the unique middle value when the data are arranged in increasing (or decreasing) order. For sets containing an even number of values, the median is the average (arithmetic mean) of the two middle values when the data are arranged in increasing (or decreasing) order. Standard Deviation It is very unlikely that a GMAT problem will ask you to calculate an exact SD. ( ) Q If each data point in a set is increased by a factor of 7, does the set's standard deviation increase, decrease, or remain constant? 48

49 S Increased by a factor of 7 means that each data point is multiplied by 7. The SD will increase. Problem Set Q {9, 12, 15, 18, 21} Which of the following pairs of numbers, when added to the set above, will increase the standard deviation of the set? A II & III I. 14, 16 II. 9, 21 III. 15, 100 S I. The numbers 14 and 16 are both very close to the mean (15). Additionally, they are closer to the mean than four of the numbers in the set, and will reduce the spread around the mean. II. The numbers 9 and 21 are relatively far away from the mean (15). Adding them to the list will increase the spread of the set and increase the standard deviation. III. While adding the number 15 to the set would actually decrease the standard deviation, the number 100 is so far away from the mean that it will greatly increase the standard deviation of the set. 49

50 Chapter 4. Consecutive Integers 等 差 数 列 性 质 : 平 均 数 = 中 位 数 =(a 1 +a n )/2 等 差 数 列 的 和 = 平 均 数 ( 或 中 位 数 ) 乘 以 个 数 ( ) 连 续 整 数 性 质 : 若 为 奇 数 个, 则 平 均 数 必 为 整 数, 总 和 必 可 被 总 个 数 整 除 ; 若 为 偶 数 个, 则 平 均 数 比 为 非 整 数, 总 和 必 不 可 被 总 个 数 整 除 ; K 个 连 续 整 数 之 积 一 定 可 以 被 k! 整 除 Consecutive integers are integers that follow one after another from a given starting point, without skipping any integers. Consecutive Even Integers: 8, 10, 12, 14 Consecutive Primes: 11, 13, 17, 19 Evenly Spaced Sets ( 等 差 数 列 ) Evenly spaced sets: These are sequences of numbers whose values go up or down by the same amount:. 50

51 Sets of consecutive multiples are special cases of evenly spaced sets: all of the values in the set are multiples of the increment:. The values increase from one to the next by 4, and each element is a multiple of 4. All evenly spaced sets are fully defined if the following three parameters are known: The smallest (first) or largest (last) number in the set The increment (always 1 for consecutive integers) The number of items in the set Properties of Evenly Spaced Sets The arithmetic mean (average) and median are equal to each other. In other words, the average of the elements in the set can be found by figuring out the median. The mean and median of the set are equal to the average of the FIRST and LAST terms. The sum of the elements in the set equals the arithmetic mean (average) number in the set times the number of items in the set. The Sum of Consecutive Integers The average of an odd number of consecutive integers (1, 2, 3, 4, 5) will always be an integer. This is because the middle number will be a single integer. On the other hand, the average of an even number of consecutive integers (1, 2, 3, 4) will never be an integer, because there is no true middle number. Q Is k 2 odd? A D (1) k- 1 is divisible by 2. (2) The sum of k consecutive integers is divisible by k. Chapter 5. Overlapping Sets The Double-Set Matrix Q of 30 integers, 15 are in set A, 22 are in set B, and 8 are in both set A and B. How many of the integers are in NEITHER set A nor set B? S 根 据 题 意 写 出 下 表 中 黑 色 部 分, 再 补 足 红 色 数 字 A Not A Total 51

52 B Not B Total The rows (columns) should correspond to the mutually exclusive options for one decision. For instance, if a problem deals with students getting either right or wrong answers on problems 1 and 2, the columns should list options for one decision problem 1 correct, problem 1 incorrect, total and the rows should list options for the other decision problem 2 correct, problem 2 incorrect, total. Overlapping Sets and Algebraic Representation Q Santa estimates that 10% of the children in the world have been good this year but do not celebrate Christmas, and that 50% of the children who celebrate Christmas have been good this year. If 40% of the children in the world have been good, what percentage of children in the world are not good and do not celebrate Christmas? S 假 设 有 100 个 人, 注 意 50% 的 意 思 是 CC 的 人 中 有 50% 的 人 HHG, 而 不 是 50% 的 人 CC 且 HHG 因 此 问 号 处 是 ? 2 Sets, 3 Choices: Still Double-Set Matrix Q of 60 children, 30 are happy, 10 are sad, and 20 are neither happy nor sad. There are 20 boys and 40 girls. If there are 6 happy boys and 4 sad girls, how many boys are neither happy nor sad? 3-Sets, 2 Choices: Venn Diagrams Q Workers are grouped by their areas of expertise and are placed on at least one team. There are 20 workers on the Marketing team, 30 on the Sales team, and 40 on the Vision team. 5 workers are on both the Marketing and Sales teams, 6 workers are on both the Sales and Vision teams, 9 workers are on both the 52

53 Marketing and Vision teams, and 4 workers are on all three teams. How many workers are there in total? S 注 意 : 要 区 分 2 set & 3 choice 和 3 set & 2 choice 的 区 别, 前 者 画 表 格, 后 者 Venn 图!! Problem Set Q 10% of all aliens are capable of intelligent thought and have more than 3 arms, and 75% of aliens with 3 arms or less are capable of intelligent thought. If 40% of all aliens are capable of intelligent thought, what percent of aliens have more than 3 arms? A 60% S x ( ) 要 理 解 第 二 句 话, 即 第 二 个 圆 圈 之 外 有 75% 的 人 落 在 第 一 个 圆 圈 内 Chapter 6. Word Problem Strategies Problem Set Q a, b, and c are integers in the set {a, b, c, 51, 85, 72}. Is the median of the set greater than 70? ( ) ( ) A A 53

54 Q A list kept at Town Hall contains that town's average daily temperature in Fahrenheit, rounded to the nearest integer, for each day of a particular completed month. Does this month have 30 or 31 days? S A (1) The median temperature is (2) The sum of the average daily temperatures is divisible by 3. Chapter 7. Extra Problem Types Scheduling Q How many days after the purchase of Product X does its standard warranty expire? (1997 is not a leap year.) S (1) When Mark purchased Product X in January 1997, the warranty did not expire until March (2) When Santos purchased Product X in May 1997, the warranty expired in May (1) Shortest possible warranty period: Jan. 31 to Mar. 1 (29 days later) Longest possible warranty period: Jan. 1 to Mar. 31 (89 days later) (2) Shortest possible warranty period: May 1 to May 2, or similar (1 day later) Longest possible warranty period: May 1 to May 31 (30 days later) Even taking both statements together, there are still two possibilities 29 days and 30 days so both statements together are still insufficient. Answer: E Problem Set Q Huey's Hip Pizza sells two sizes of square pizzas: a small pizza that measures 10 inches on a side and costs $10, and a large pizza that measures 15 inches on a side and costs $20. If two friends go to Huey's with $30 apiece, how many more square inches of pizza can they buy if they pool their money than if they each purchase pizza alone? S 注 意 是 square!! A 25 square inches 54

55 Q Each senior in a college course wrote a thesis. The lengths, in pages, of those seniors' theses are summarized in the graph above. a. What is the least possible number of seniors whose theses were within six pages of the median length? b. What is the greatest possible number of seniors whose theses were within six pages of the median length? S 横 坐 标 为 论 文 张 数, 纵 坐 标 为 人 数 共 20 人 中 位 数 是 第 10 个 人 和 第 11 个 人 张 数 ( 都 在 范 围 ) 的 平 均 值, 这 两 个 人 一 定 满 足 a 的 条 件 a. 2: Since the tenth- and eleventh-longest papers must satisfy the criterion, place them as close together as possible (to leave room to manipulate the other lengths): say 28 pages apiece. If the other four 20- to 29-page papers are each 20 or 21 pages, and all seven 30- to 39-page papers are each 35 pages or more, then only the tenth- and eleventh-longest papers are within six pages of the mean. b. 17: If the tenth-longest paper is 25 pages and the eleventh-longest paper is 24 pages, then the median length is 24.5 pages, so all of the 20- to 29-page papers are within six pages of the median. If each of the four 10- to 19-page papers is 19 pages long and each of the seven 30- to 39-page papers is 30 pages long, then all eleven of those papers will also be within the desired range. Chapter 8. Extra Consecutive Integers Products of Consecutive Integers and Divisibility The product of k consecutive integers is always divisible by k factorial (k!) Sums of Consecutive Integers and Divisibility For any set of consecutive integers with an ODD number of items, the sum of all the integers is ALWAYS a multiple of the number of items. 55

56 This is because the sum equals the average times the number of items. For any set of consecutive integers with an EVEN number of items, the sum of all the items is NEVER a multiple of the number of items. This is because the sum equals the average times the number of items. For an even number of integers, the average is never an integer. Consecutive Integers and Divisibility Q If x is an even integer, is x(x + 1)(x + 2) divisible by 4? S 因 为 x 是 偶 数,x+2 也 是 偶 数, 所 以 可 以 被 4 整 除 Problem Set Q Is the sum of the integers from 54 to 153, inclusive, divisible by 100? A NO, 因 为 有 100 个 数, 偶 数 个 连 续 整 数 一 定 不 能 被 该 偶 数 整 除 Q Is the average of n consecutive integers equal to 1? (1) n is even. (2) If S is the sum of the n consecutive integers, then 0 < S < n. A D S (1) 偶 数 个 连 续 整 数 的 平 均 值 一 定 不 是 整 数, 所 以 一 定 不 为 1 (2) 不 等 式 同 时 除 以 n, S/n<1, 而 S/n 就 是 平 均 值, 故 一 定 不 能 为 1 56