# Primes. Name Period Number Theory

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1 Primes Name Period A Prime Number is a whole number whose only factors are 1 and itself. To find all of the prime numbers between 1 and 100, complete the following exercise: 1. Cross out 1 by Shading in the box completely. 1 is neither prime nor composite. It has only 1 factor - itself.. Use a forward Slash \ to cross out all multiples of, starting with 4. is the first prime number. 3. Use a backward Slash / to cross out all multiples of 3 starting with Multiples of 4 have been crossed out already when we did #. 5. Draw a Square on all multiples of 5 starting with is prime. 6. Multiples of 6 should be X d already from # and #3. 7. Circle all multiples of 7 starting with is prime. 8. Multiples of 8 were crossed out already when we did #. 9. Multiples of 9 were crossed out already when we did # Multiples of 10 were crossed out when we did # and #5. All of the remaining numbers are prime. How many prime numbers are left between 1 and 100? Answer: Use your chart for help. Is 51 prime? If not, what are its factors? Is 59 prime? If not, what are its factors? Is 87 prime? If not, what are its factors? Is 91 prime? If not, what are its factors?

2 Divisibility Rules Name Period There are some easy tricks you can use to determine if a number is divisible by, 3, 4, 5, 6, 8, 9 and 10. A number is divisible by: - if it is even. 3 - if the sum of its digits is divisible by if the number formed by the last digits is divisible by 4. (ask me why this works) 5 - if the ones digit is 5 or if it is divisible by AND 3. (All even multiples of 3.) 7 - there is no good trick for if the number formed by the last 3 digits is divisible by 8. (ask me why this works) 9 - if the sum of the digits is divisible by if the last digit is a 0. 11: We will learn this trick next. It is more complicated. Practice: Write yes or no in each blank. Determine whether 1,408 is divisible by: Determine whether 1,345,866 is divisible by: Determine whether,,5 is divisible by:

3 Trickier Divisibility Problems Examples: 1. What is the smallest 4-digit number that is divisible by, 3, 4, 5, 6, 8, 9, and 10? Reasoning: The number must end in zero. Lets assume that to be the smallest it should start with a 1. Since the digits must add up to 9, the last three digits must add up to 8. 1,080 is the smallest four-digit integer divisible by 8 and 9. A second method will be explained in the first practice problem below.. Using only 1s and s, what is the smallest integer you can create which is divisible by both 3 and 8? Reasoning: The last digit has to be a. Since 1 and don t work, we need a 3-digit number that is divisible by 8. 11/8 = 14, but it is not divisible by 3. Unfortunately, 1, 1, and are not divisible by 8 so we must go to a 4-digit number that ends in 11 and whose digits are a multiple of 3. The only 4-digit number that works is,11 (we want the sum of the digits to be 6) so it is the smallest. Practice: 1. What is the smallest positive integer that is divisible by, 3, 4, 5, 6, 8, 9, and 10? (There is a good way to do this without guess-and-check).. The digits of a number are all 8s, and it is divisible by 9. What is the smallest positive integer that fits this description? 3. What is the smallest positive integer that is divisible by and 3 that consists entirely of s and 3s, and has at least one of each? 4. What is the smallest 5-digit integer divisible by both 8 and 9? ,888,888 3., ,008

4 Divisibility Rule: Eleven The divisibility rule for 11 is seldom taught in regular classes. Practice: First, take a moment to multiply several numbers by 11: abcd x11 x11 x11 You should see some patterns with the digits. In the final example, the digits become: a a+b b+c c+d d If you add the alternating digits you get the same result. a a+b b+c c+d d To find out if a number is divisible by eleven: Sum the alternating digits. Subtract these two numbers. If the result is zero or is divisible by 11, the number is divisible by 11. Examples: Determine if each number is divisible by 11 without a calculator: , , ,91 Practice: Determine if each number is divisible by 11 without a calculator: 1. 3, , , ,768 Harder Practice: Solve each without a calculator. 1. What digit could fill-in the blank if 89_43 is divisible by 11?. What five-digit multiple of 11 consists entirely of s and 3s? 3. What is the largest five-digit multiple of 11? 4. What is the remainder when you divide 1,34,567 by 11?

5 Divisibility Practice Practice: Solve each using what you have learned about divisibility. Name Period 1. What digit could be used to fill in the blank and make the following number divisible by both 3 and 8? 45,_8. What is the smallest three-digit prime? 3. How many multiples of 3 less than 1,000 use only the digits and/or is divisible by both 8 and 9. How many integers less than 360 are also divisible by both 8 and 9? (hint: First find the smallest integer that is divisible by both 8 and 9.) 5. A three-digit integer is divisible by 9. If I subtract the tens digit from the hundreds digit, I get the ones digit. What is the largest number that meets these conditions? 6. There are two ways that the digits 1,, 3, and 4 be arranged to create a four-digit multiple of 8. Find them both. 7. Consecutive integers are placed in order to form a three-digit integer. The integer will ALWAYS be divisible by what prime number? 8. For the number ABC, each distinct letter represents a different digit. If ABC, CAB, and BCA are all divisible by 6 and 9, find the value of ABC + CAB + BCA. 9. What is the largest seven-digit number that contains each of the digits 1 through 7 and has the property that the sum of any two consecutive digits is a prime number? (source: MATHCOUNTS 000 National Team Round)

6 Quiz: Divisibility Solve: Name Period 1. How many of the following integers are factors of 1,345?, 3, 4, 5, 6, 8, 9, What number could fill-in the blank to make 56,74_ divisible by both 3 and 4?. 3. How many prime numbers are greater than 0 but less than 40? Using only the digits 1,, and 3 with at least one of each, what is the smallest integer that can be created which is divisible by 8 and 9? Using only the digit, how many s must be used to create an integer that is divisible by both 9 and 11? (Example: Using four s:, is divisible by 11 but not 9.) What is the remainder when 456,654,465,645 is divided by 6? The digits 0,3,6, and 9 are used to create the smallest integer that is divisible by, 3, 4, 5, 6, 8, 9, 10, and 11. What is this integer? 7.

7 Quiz: Divisibility Solve: Name Period 8. The digits 5, 6, and 7 are arranged to create a three-digit number. Which of the following cannot be a factor of the number formed? (There may be more than one answer, list all that apply.) Using two 5s and two 6s, it is possible to create four 4-digit numbers which are divisible by 11. What is the sum of these four numbers? Each of the digits 0-9 is used exactly once to create a ten-digit integer. Find one of the many ten-digit numbers which uses each digit once and is divisible by 8, 9, 10, and

8 Divisibility Combo Rules: Each player gets 6 cards, each with a single digit. Each hand, players goal is to find a number using at least 3 of their 6 digits which meets the divisibility requirements. Example: You hold 1,4,4,5,7 and 8. Goal: Divisibility by 3 and 5. Possible answer: 8,415 4,875 etc. Example: You hold 1,,3,3,8 and 9. Goal: Divisibility by 3 and 8. Can you find a solution? Sometimes it will be impossible using the cards you have. After a player gets three combos, (s)he wins the round and everyone gets a new hand. Create a 3+ Digit Number that is Divisible by: 6 and 3 3 and 4 5 and 6 and 11 3 and 11 8 and 9 4 and and 8 3 and 8 4 and and 5 and 11 5 and 9 3 and 10 and 3 and 5

10 Testing for Primes Testing to see if a number is prime: If we want to know if 401 is prime, do we need to test to see if the following numbers are factors: 6? 7? 10? 13? 0? 3? Is 401 prime? How can you tell? To determine whether a number n is prime: Check for divisibility by primes < n starting from least to greatest. Think: You do not need to check composites like 6 and 14 because if 6 were a factor, and 3 would be factors. If 14 were a factor, 7 would also be a factor. You do not need to check primes greater than the square root because they would be multiplied by a number less than the square root (which we would have already checked). Example: Is 181 prime? Check in your head:, 3, 5, and 11. Check 7 and 13 on paper if you need to. Make sure you understand why you do not need to check numbers like 4 and 17 as factors. Practice: Answer each (you wil need a calculator). If a number is composite, write two numbers whose product is the number. 1. Is 391 prime?. Is 87 prime? 3. Is 503 prime? 4. The number 13 is prime, and when its digits are reversed, 31 is also prime. In addition to 13 and 31, five other -digit primes satisfy this condition. What are they? Challenge: What is the smallest 4-digit prime? 1. 17x3. 7x41 3. Prime , 17-71, C. 1001=7x11x =17x =19x is :prime Notice that you do not need to check 7, 11, 13, 17, or 19 (for 1009) since they are factors of 1001, 1003, and 1007.

11 A Brief Intro to Exponents One of the things that will come up frequently in this class which you all must be familiar with is the use of exponents. Base: The repeated factor in a power. In the expression n³, n is the base. Exponent: Represents the number of times a factor is being multiplied. In the expression n³, the ³ is the exponent. The expression The expression The expression 3 5 means that you multiply x means that you multiply x x x x x 4 (ab) means that you multiply ( ab)( ab)( ab)( ab) Practice: Write-out without using exponents: n 3. 5 ( 3n) a Practice: Write using exponents. Ex a a a a n n n 3. 5 Practice: Evaluate. (solve) Ex

12 Fundamental Theorem of Arithmetic Today we will learn something so crucial to number theory and mathematics that it is actually called the FUNDAMENTAL THEOREM OF ARITHMETIC! The theorem states that Every Positive Integer has a Unique Prime Factorization. Example: 5,544 = 3 x 3 x 7 x 11 and there is no other way to factor 5,544 into a product of primes. To find the prime factorization of a number, it is often easiest to create a factor tree. If you remember your divisibility rules this should be easy: Examples: Factor each (try without a calculator) ,518 Practice: Write the prime factorization of each number below using exponents and placing the prime factors in order from least to greatest Practice: Write the prime factorization of each number below. These three numbers all have something in common. Can you tell what it is? The numbers above are all perfect squares. If a number is a perfect square, each prime factor will have an even exponent. Example: (3 5) What perfect squares are represented below?

13 Fundamental Counting Principle Another important Fundamental in mathematics is The Fundamental Counting Principle: See Probability Unit for worksheet

14 Counting Factors One way to count the number of factors (divisors) that a number has is to list them. Examples: 40: 96: 196: When you list factors, list them in pairs and go from least to greatest. There is a nice relationship between the prime factorization of a number and the number of factors (divisors) that it has. Example: Look at the prime factorization of 40: Every factor of 40 is a combination of s and 5s If you are only asked HOW MANY factors a number has, there is an easy shortcut that involves the prime factorization (maybe you can recognize what this shortcut is by looking at the example above): Example: How many factors does 56 have? First find the prime factorization Each factor can have either 0, 1,, or 3 twos and 0 or 1 seven. This means that 56 has 4 choices for the number of twos and two choices for the number of sevens in each of its factors, for a total of 4x=8 factors. They are 1,, 4, 7, 8, 14, 8, and 56 This is a great trick and we will practice more with it.

15 Counting Factors Name Period Practice: How many factors (divisors) does each number have? (do not list them) How many factors does 600 have? Apply what you know: 6. What is the smallest positive integer that has exactly 6 factors? Think: What does the prime factorization of a number with 6 factors look like? 7. What is the smallest positive integer that has exactly 10 factors? 8. What is a characteristic of all numbers with exactly two positive factors? 9. What is a characteristic of all positive integers that have an odd number of factors? 10. What is the prime factorization of the smallest positive integer that has exactly 31 factors? Challenge. Five -digit positive integers have exactly 1 factors. List them.

16 Counting Factors Trickier factor counting: Examples: 1. How many odd factors does 40 have? Reasoning: First look at the prime factorization of 40: Any number times is even, so the odd factors are the ones that have no twos. I can only use the 3s and 5s for a total of x=4 factors: 1, 3, 5, and 15.. How many perfect squares are factors of 360? Reasoning: Look again at the number s prime factorization: We need to find factors that have two twos, two threes, or both. 5 1 cannot be a factor of a perfect square. Our perfect squares are 0 =1, =x=4, 3 =3x3=9, and x3 =xx3x3=36. Finding the prime factorization of a number quickly is the key to MANY number theory problems. Practice: 1. How many perfect squares are factors of 400?. How many even factors does 10 have? 3. If a number n has 7 factors, how many factors does n have? 4. The number n is a multiple of 7 and has five factors. How many factors does 3n have? Challenge: The number p is a multiple of 6 and has 9 factors. How many factors does 10p have? 1. 6: 1, 4, 16, 5, 100, :, 6, 10, 14, 30, 4, 70, (consider n 6 and n 1 ) 4. The original number must be 7 4 so the new number 3x7 4 has 10 factors C. 4 (p=xx3x3)

17 Counting Factors Name Period Basics: How many factors (divisors) does each number have? (do not list them) Basics: How many ODD factors (divisors) does each number have? (do not list them) , 600 Basics: How many EVEN factors (divisors) does each number have? (do not list them) , 00 Basics: How many Perfect Square factors (divisors) does each number have? List them , 96 Apply what you know: 9. How many two-digit numbers have EXACTLY three factors? 10. What is the smallest positive integer that has 18 factors?

18 Review: Divisibility 100. Find the smallest positive integer greater than 90,000 that is divisible by What positive integer could fill-in the blank to make the following divisible by six? 8,76_ 300. What positive integer could fill-in both blanks to make the following divisible by both 4 and 9? 8_,8_ What is the smallest 5-digit integer that is divisible by, 3, and 5? 500. What is the least positive integer that is divisible by 9 using only the digits 3 and 4, and having at least one of each? 600. What is the least positive integer that is divisible by 8, 9, 10, and 11? Primes and Prime Factorization (calculators o.k.) 100. What is the prime factorization of 43? 00. What is the prime factorization of 35? 300. List all of the primes less than 1000 that use only one digit. (Start with 11.) 400. What is the prime factorization of 33? is prime. What is the next prime integer after 911? 600. What is the prime factorization of 1,681? How many Factors does each have? , , 750

19 Practice Quiz: Factors & Divis. Solve: 1. How many of the following integers are factors of 888?, 3, 4, 5, 6, 8, 9, 10. Name Period. What number could fill-in the blank to make 8_,045 divisible by 11? What is the smallest prime number that is greater than 150?. 4. What is the Prime Factorization of 64 (written with exponents)? How many factors does 600 have? The digits of a 5-digit positive integer are 1s, s, and 3s with at least one of each. What is the smallest such integer that is divisible by both 8 and 9? What is the smallest positive integer that has exactly 10 factors? 6. 7.

20 Practice Quiz: Factors & Divis. Solve: 8. Distinct positive integers a and b have 5 and 6 factors respectively. What is the smallest possible product ab if a and b do not have any factors in common greater than 1? Name Period How many even factors does 990 have? How many 3-digit integers have exactly 3 factors? 10.

21 Quiz: Factors & Divisibility Solve: Name Period 1. How many of the following integers are factors of 9,16?, 3, 4, 5, 6, 8, 9, What number could fill-in the blank to make 13,_45 divisible by 11?. 3. What is largest -digit prime number? What is the Prime Factorization of 378 (written with exponents)? 5. How many factors does 350 have? A positive 6-digit integer has two different digits. What is the smallest such integer that is divisible by both 5 and 6? What is the smallest counting number (positive integer) that has exactly 8 factors? 6. 7.

22 Quiz: Factors & Divisibility Solve: Name Period 8. How many of the factors of 900 have exactly 18 factors? What is the smallest odd counting number that has exactly 1 factors? How many primes are there between 100 and 144? 10.

23 Practice: Challenge Questions Solve: 1. What is the smallest odd integer that has 8 factors? Name Period. How many perfect squares are factors of 1,31? Two four-digit primes use only the digits 3 and 5. Find them both.. 4. What is the smallest prime factor of 1,517? What is the maximum number of factors that any three-digit number has? 6. Which two three-digit multiples of 11 have exactly 10 factors? How many different rectangles of integer length and width have an area of 1,00cm? The year 1,849 was the last year with 3 factors. What is the next calendar year with 3 factors? 7. 8.

24 Product of the Factors (Divisors) Now that you can count the number of factors of any natural number (positive integer) quickly, lets find a quick way to find the product of those factors. Begin again by making a list of all factors of 10 and 144: 10: 144: Using the method we used before, it is easy to see that the factors come in pairs (except for perfect squares, we will discuss this in a moment). Fill in the blanks to complete the sentences below: 10 has factors, which can be divided into pairs, and the product of each pair is. Therefore, the product of all of the pairs is to the power. 144 has factors. Since each pair of factors has a product of, we can replace every factor with the number without changing the product. Therefore, the product of the factors of 144 is to the power. If you know a little about fractional exponents, the products of the factors can be written as: and Using the logical statements above may be more helpful to most of you. Examples: Find the product of the factors of each Practice: Find the product of the factors of each

25 Product of the Factors Practice Solve: 1. The product of the factors of 50 is. Place in order from least to greatest: a. The product of the factors of 15. b. The product of the factors of 5. c. The product of the factors of 1. x 50. What is x? is a prime number. What is the product of the factors of 199? Name Period A number x has n factors. What is the product of the factors of x in terms of x and n? Write the prime factorization for the product of the factors of The product of the factors of 30 is equal to x x x 3 5. What is x? The factors of the number one can be expressed as 1. The product of what number s factors can be expressed as3? 7. Challenge: The product of the factors of 36 is divided by the product of the factors of 54. Write the result as a fraction in simplest form. C.

26 Sum of the Factors Example: Consider the following problem: Find the sum of the factors of 7. This appears to be a very tedious task. We could list all of the factors (you should know now that there are 1 of them) and then add them up. Like most of what we have learned in, there is a shortcut. The shortcut requires that you know some Algebra, specifically how to distribute. We will learn to distribute with integers. This will be review if you have studied Algebra. Begin by finding the area of each rectangle below There are two ways to find the area: 1. Find the length of both sides, and multiply to get the entire area.. Find the area of the pieces and add them together. Distribution is the same as finding the pieces and adding them. 6(9 4) 6(9) 6(4) ( 5 4)(7 3) 5(7) 5(3) 4(7) 4(3) More Practice: Do these the long way! Check your work using the short way. ( 5 )(7 3) (8 3)(5 4 )

27 Sum of the Factors Now, apply this to finding the sum of the factors of a positive integer: Example: What is the sum of the factors of 0? (this one is easy to check) We can get all of the factors using the trick we just learned with distribution. Multiply the sum of the powers of by the sum of the powers of 5: ( )(5 5 ) First, lets make it easier to read. ( 1 4)(1 5) Distribute. This gives us every factor! is the sum of all the factors. Of course, we don t need to use the long way! ( 1 4)(1 5) (7)(6) 4 This is more important with bigger numbers! Examples: Find the sum of the factors of each Practice: What is the sum of the factors of each? Check this one by writing all the factors out

28 Sum of the Factors Solve: Name Period 1. What is the sum of the factors of: a. 50 b. 405 c. 10. Which single-digit integer has a greater factor sum: 6, 8, or 9? 3. Find the sum of the factors of each number below and look for a pattern to determine the sum of the factors of 30. a. 64 b. 18 c The prime factorization of a number is n 3. The sum its factors is 400. What is n? 5. What is the average of all the factors of 144? 6. 6 n has a factor sum that is greater than 1,000. What is the smallest integer value of n? 7. The sum of the factors of x 3 x 5 x 7 x 11 is divided by the number of factors in 11 x 3 7 x 5 5 x 7 3 x 11. What is the result?

29 Perfect/Abundant/Deficient Solve: 1. What is the sum of the factors of: a. 54 b. 300 c Name Period. The numbers 6 and 8 are the two smallest perfect numbers. Find the property that makes these numbers perfect. Is 496 perfect? A perfect number is a number in which the sum of its proper divisors (all of its factors, not including itself) is equal to the number itself (this is the same as saying the sum of the factors is twice the number) 3. There are very few perfect numbers. Most numbers are either abundant or deficient. If 30 is an abundant number, and 35 is considered deficient, do you think that 100 is abundant or deficient? Explain. 4. Are all prime numbers abundant or deficient? Explain why. 5. Are all multiples of 6 (greater than 6) abundant or deficient? Explain why. 6. Are all powers of abundant or deficient? Try to explain why.

30 GCF (also called GCD) and LCM GCF stands for Greatest Common Factor The Greatest Common Factor (also called Greatest Common Divisor) for a pair or set of integers is the largest number that is a factor of each. LCM stands for Least Common Multiple The Least Common Multiple for a pair or set of integers is the smallest integer for which each number is a factor. GCF and LCM should be taught extensively in your regular math class, so we will only have a short lesson on GCF and LCM (we will learn a technique or two that is different than what is taught in most classrooms). Sometimes The GCF and LCM are obvious: Practice: What is the GCF for the following pairs of numbers? and and and 54 (careful!) Practice: What is the LCM for the following pairs of numbers? 1. 4 and and and 40 (careful!) It is not always so easy. What if you were asked to find the GCF and LCM of 84 and you may not see the answer in your head. Venn Diagrams are a great way to solve GCF and LCM problems. Example: Use a Venn diagram to find the GCF and LCM between 84 and Example: Use a Venn diagram to find the GCF and LCM for 75 and 105: Practice: Use a Venn diagram to find the GCF and LCM for each and and , 4, and 105

31 GCF and LCM Review: Find the GCF and LCM for 45 and 105. Now, multiply the GCF by the LCM. Finally, multiply the original numbers. Can you explain why this works? (Try looking at the Venn diagram.) The product of a pair of numbers is equal to the product of their GCF and LCM. Before we move on, lets try using our technique with three numbers: Example: Find the GCF and LCM of 30, 45, and 50 using a Venn Diagram. Practice: Find the GCF and LCM of 56, 16, and 10 using a Venn Diagram. Note: There are slightly quicker methods that you may know already, or will learn which can be used for finding the GCF and LCM. I like this one because it illustrates very clearly the concepts of shared/common factors and helps bright students understand the other methods. Trickier LCM problems: 1. You are buying cups, plates, and napkins for a picnic. Cups come in packs of 4, plates come in packs of 30, and napkins come in packs of 100. How many packs of napkins will you need to buy if you want to buy the same number of cups, plates, and napkins?. Janice and Kiera begin jogging around a track, starting at the finish line and going the same direction. Janice completes a lap every 78 seconds, while Kiera takes 90 seconds. At the end of their workout, they cross the finish line together in a whole number of minutes. a. What is the minimum number of minutes that they could have run for? b. How many more laps did Janice run than Kiera? 3. Ken gets his hair cut every 0 days. Larry gets his cut every 6 days. Ken and Larry get their hair cut on the same Tuesday. What day of the week is it the next time they get their hair cut on the same day?

32 4! = 4 Factorials You may have seen an exclamation point in a math problem at some point and wondered, What is so exciting about that number? No, 4! does not mean FOUR!!! That exclamation point is actually not there to represent excitement or volume, it is factorial notation. n! is equal to the product of all integers less than or equal to n. Examples make this simple: 5! ! ,68,800 15! ,307,674,368,000 You can see that factorials can get big fast! You will not be asked to simply compute large factorials, but there are many interesting problems involving factorials that we will review. First, some simple arithmetic and Algebra: Examples: 5! 4! 1. 5! ! 5! 3. Harder Practice Problems: 1! 19! 1. 6!. 60 5! 4! 3! 3.

33 Factorials We will solve many factorial problems using the same techniques we have learned in this unit involving prime factorization. Examples: 1. Write the prime factorization of 5! using exponents.. How many factors does 5! have? 3. Find the sum of the factors of 5!. 4. Find the product of the factors of 5!. More Examples: 1. What is the GCF of 5! and 11!?. Find the largest prime factor of 15!. 3. What is the LCM of 5! and (3!)? 4. What is the GCF of 6! and 1 3? More Problems Involving Factorials: 1. Find the largest power of 3 that divides (is a factor of) 9!.. Find the smallest possible value of n if n! is divisible by When 100! is written as an integer, how many zeroes does it end with? 4. In the prime factorization of 1000!, what is the power of 11?

34 Factorials Practice Practice: 1. Write the prime factorization of 7! using exponents. Name Period. How many factors does 6! have? 3. What is the LCM of 6! and 600? 4. Find the largest prime factor of 100!. 5. Find the sum of the factors of 7! (hint: use #1) 6. Find the largest power of 6 that divides 18!. 7. Find the smallest possible value of n if n! is divisible by When 15! is written as an integer, how many zeroes does it end with?

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