Black Problems - Prime Factorization, Greatest Common Factor and Simplifying Fractions

Transcription

1 Black Problems Prime Factorization, Greatest Common Factor and Simplifying Fractions A natural number n, such that n >, can t be written as the sum of two more consecutive odd numbers if and only if n is prime or n is twice an odd number Of the ten natural numbers 0 through 9, how many can t be written as the sum of two or more consecutive odd numbers? Count prime dates A prime day has a month and day which are both prime Thus 5/3 is a prime date How many prime dates are there in the year 000? [Dec 998 (3)] 3 What is the positive difference between the greatest and least prime factors of 000? 4 What is the greatest prime factor of ? 5 What is the least whole number value of x such that f(x) x x is not prime? 6 The 6 letters of the alphabet are assigned prime number values in consecutive order beginning with The product of the values of the letters of a common math term is 595,034 What is the term? What is the only prime number which is the sum of four consecutive prime numbers? 8 What is the smallest prime factor of 8? 9 What is the least positive fraction whose numerator is two less than a perfect square and whose denominator is one more than the same perfect square? 0 Between 3:00 pm and 4:00 pm, for what fractional part of the hour does the digit appear on a hour digital clock that shows hours and minutes? Express your answer as a common fraction Simplify: 3a 5m a 5am 6m a m a 3m Think about all of the positive common (reduced) fractions with a denominator of or less How many of these common fractions have a value less than? 3 Supposed we replace each x in the expression x with the expression x x x What is the value of the resulting expression when x 4? Express your answer as a 5 common fraction

2 4 ABCD is a rectangle, AB 6, BC 4, EFGH is a parallelogram, AE/BE /, and BF/FC /3 What is the ratio of the area of parallelogram EFGH to the area of rectangle ABCD? Express your answer as a common fraction A E B F H D G C 5 Both 5 and 9 are prime numbers that can be written as sums of squares: 5 and 9 5 Which of the following numbers has the same properties: 3,, 53, 3, 69? Black Solutions The primes from 0 to 9 are 3 and 9 The numbers that are twice an odd number are and 6 These 4 natural numbers can t be written as the sum of two or more consecutive odd numbers Count prime dates Fiftythree Prime months are, 3, 5,, and ; prime days are, 3, 5,,, 3,, 9, 3, 9, and 3 If all prime months had 3 days, there would be 55 prime dates But February in 000 has 9, and November has 30 So there are 53 prime dates 3 The prime factorization of 000 is 4 x 5 3 The greatest prime factor is 5, and the least prime factor is The difference between these two numbers is Factoring gives ( ) (308) Further factorization gives 5 00 x 00 x (3 x 3 x 9) The greatest prime factor is 9 5 Value of the function can be checked with a chart x x x When x 0, the value of the function is x x 0 0 This is the first value of the function which is not prime, so the answer is 6 Since the product consists of values that are primes, find the prime factorization of 595,034 When factoring only try prime values, because if a prime doesn t work then no multiple of that prime will work The number is even, so quite obviously is a factor: 595,034 x 9,5 Trying to find factors of 9,5 is a bit more difficult; 3 doesn t work, 5 doesn t work, doesn t work The first to work is : 9,5 x,04 Using a calculator then yields that,04 x 3 x 43 The prime factors are,,, 3 and 43, and these primes correspond to the letters a, e, g, l, and n, which can be rearranged to spell angle Every prime number except is odd The sum of any four odd numbers is even Thus, the only way to get an odd sum for four primes is if one of the primes is The only set of consecutive primes that

3 contains is the set {, 3, 5, } The sum of these four primes is Thus, is the only prime that can be written as the sum of four consecutive primes The perfect squares around which our fraction will be built are, 4, 9, 6, etc We cannot use since the numerator will become negative when we subtract Let s try 4 The numerator is two less, which is 4, and the denominator is one more, which is 4 5 The fraction is /5 To be sure, we check that the next possibility, /0, is clearly more than /5, as is 4/, and the rest, which get closer and closer to a value of In each case out numerator is three less than our denominator In terms of a pizza, for instance, the smaller the denominator (or number of total pizza pieces), the more of an impact it is to decrease the numerator by three (take away three pizza pieces), so we want the smallest possible denominator 0 The digit will appear on the digital clock for a full ten minutes from 3:0 through 3:9 and for five more full minutes at 3:0, 3:, 3:3, 3:4, and 3:5 That s 5 minutes in all, which is /4 of the hour 3a 5m a 5am 6m a m 3a 5m a 3m (a m)(a 3m) a m a 3m 3a 5m a 3m (a m) 3a 5m a 3m a 4m (a m)(a 3m) (a m)(a 3m) a 4m (a m)(a 3m) (a m) (a m)(a 3m) a 3m A positive fraction can be represented as a lattice point on a grid using the ordered pair (denominator, numerator) One fraction is less than another fraction if the segment joining it to the origin (0, 0) is below the segment joining the other fraction to the origin In Figure we have drawn the segment connecting to the origin with a solid line Every Figure other common fraction with a denominator of will fall along the line x The segments representing the positive fractions with a denominator of and a value less than are drawn with dotted lines We can see there are only six dotted lines, so there are six positive fractions with a denominator of that are less than We can then go to a denominator of 0 and see how many of these fractions work In Figure other words, how many lattice points are along the line x 0 and below the segment representing Notice that when we eventually draw in the corresponding segment for the fraction 5, if will fall along the exact same segment that was drawn for 0, so we ll know not to count it again This is how we will ensure we are counting only common fractions We can see from Figure that this is going to get messy and difficult to count, so it might not be the best method However, without drawing in all of the lines, we can get good idea from the lattice points about which fractions are less than Careful graphing and counting yields an answer of 6 In Figure 3 we see there are 36 lattice points below the segment representing Figure 3 (and above the xaxis) but the 0 open circle lattice points do not represent common fractions The answer is fractions

4 3 Substituting the value 4/5 into expression (x )/(x ), we get (4/5 )(4/5 ) (9/5)/(/5) 9 Now we substitute this value again into the expression (x )/(x ), which gives us (9 )/(9 ) ( 8)/(0) 4/5 4 The area of rectangle ABCD is 6 x 4 4 square units Triangles AEH and CFG together make a rectangle that is 3 x 4 square units Likewise, triangles DGH and BEF together make a rectangle that is x square units Subtracting and from 4, we find that parallelogram EFGH has an area of 0 square units The ratio of the area of parallelogram EFGH to rectangle ABCD is Notice that 3 is prime We must only look at values that are less than 3 By trial and error we find that 3 6 4

5 Bibliography Information Teachers attempted to cite the sources for the problems included in this problem set In some cases, sources were not known Problems Bibliography Information Collier, C Patrick Menu Collection Problems Adapted from Mathematics Teaching in the Middle School New York: National Council of Teachers of Mathematics, 000 Print, 3 4 Math Counts ( 5 Cook, Allen, and Natalia Romalis Content Area Mathematics for Secondary Teachers The Problem Solver New York: Christopher Gordon, Inc, 006