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NAME ________________________ Date Blue Problems Prime Factorization, Greatest Common Factor and Simplifying Fractions 1. The number n is a prime number between 20 and 30. If you divide n by 8, the remainder is 5. What is the value of n? 2. Five consecutive two-digit positive integers, each less than 30, are not prime. What is the largest of these five integers? 3. What is the sum of the three distinct prime factors of 47,432? 4. What is the tens digit of the product of the first six prime numbers? 5. What is the greatest prime factor of 3105? 6. In a certain code, each of the 26 letters of the English alphabet is represented by a number (A=1, B=2, C=3,... Z=26). A word is then encoded by multiplying the numbers that represent its letters. For example, CAT is encoded by 3* 1* 20 = 60, MATH is encoded by 13*1*20*8 = 2080. Find a word that would be encoded as 7560 and explain how you found it. Could there be other words? Explain why or why not. 7. Apples. At harvest time, the orchards of Mr. MacIntosh, Mr. Jonathan, and Mr. Delicious had yielded 314,827 apples, 1,199,533 apples, and 683,786 apples, respectively. While having lunch with Jonathan the following Sunday, MacIntosh mentioned the number of apples he would have left over if he divided his harvest equally among all the apple dealers. “Why don’t you sell those extra apples to me,” suggested Jonathan, “and then I’ll be able to divide my apples equally among all the dealers.” “Sorry, said MacIntosh, “but Mr. Delicious made the same suggestion for the same reason, and I’ve already accepted his offer.” How many apple dealers are there? 8. What is the positive difference between the two largest prime factors of 159,137? 9. Last Prime Date. If both the month and the day are prime numbers, then consider this date to be a prime date. For example, July 19 (7/19) is a prime date. What are the first and last prime dates of the year? 10. What number can be subtracted from both the numerator and denominator of 19/24 so that the resulting fraction will be equivalent to 3/4? 11. What fraction of the eleven letters in the word “MISSISSIPPI” are I’s? Express your answer as a common fraction. 12. The fraction x does not change its value when 3 is added to both its numerator 5 and its denominator. What is the value of x? 13. A fraction is equivalent to 3/5. Its denominator is 60 more than its numerator. What is the numerator of this fraction? 14. Find two pairs of variable expressions that have 24xy2 as their greatest common factor. 15. The sum of the numerator and denominator of fraction is 160. The fraction is equivalent to 3 . What is the original fraction? 7 Blue Solutions 1. The primes between 20 and 30 are 23 and 29. If we divide 23 by 8, we get a remainder of 7. If we divided 29 by 8, we get a remainder of 5. The value of n must be 29. 2. This problem is asking about two-digit numbers that are not prime, in other words, they will each be composite numbers. We are given an upper limit of 30, so all of these numbers will be in the teens or twenties. Let’s go ahead and list them out, circling the numbers that are not prime: 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 5 non-primes The largest of the five consecutive non-prime integers is 28. 3. The prime factorization of 47,432 is 23x72x112. The sum of the three prime factors is 2 + 7 + 11 = 20. 4. The first six prime numbers are 2, 3, 5, 7, 11 and 13. The product of the three primes 7, 11 and 13 is the very special number 1001. This is helpful in determining the product of all the numbers. Since 2 3 5 = 30 and 7 11 13 = 1001, then the product of all six primes is 30 1001 = 30,030. The tens digit is 3. 5. Since the digits of 3105 have a sum of 3 + 1 + 0 + 5 = 9, 3105 is divisible by 9. In fact, 3105 = 3 2 x 345. However, the digits of 345 have a sum of 3 + 4 + 5 = 12, so 345 must be divisible by 3. This implies that 3105 = 33 x 115. Finally, since 115 ends in 5, it is divisible by 5, giving 3105 = 3 3 x 5 x 23. The greatest prime factor of 3105 is 23. 6. LINE, ALIEN, REGAL, BORN, LARGE, BARON are only some of the many. We began to solve the equation by making a factor tree. We factored the number 7560 until we received four numbers that translated into letters, which we formed into a real word. This is an example of our factor tree: Numbers used: Translation of numbers: Letters unscrambled: 7560 /\ 20 378 /\ / \ 4 5 18 21 DE RU RU DE 7. Apples. The number of dealers must be a divisor of 314,827 + 1,199,533 = 1.514.360, and of 314,827 + 683,786 = 998,613. The greatest common divisor of these two numbers is 131. Since 131 is a prime, there are 131 dealers. 8. Prime factoring a number is arguably the most difficult thing to do in mathematics. There’s no quick way to do it. But, the divisibility rules will show that no prime less than 11 will divide the number in the problem. Using the divisibility rules is a good way to get started. The divisibility rule for 2 says that a number is divisible by 2 if the last digit is even. The last digit is 7, so the number is not divisible by 2. The divisibility rule for 3 says that a number is divisible by 3 if the sum of the digits is divisible by 3. Since 1 + 5 + 9 + 1 + 3 + 7 = 26 is not a multiple of 3, then 159,137 is not divisible by 3, either. A number divisible by 5 has units digit 0 or 5, so 159,137 is not divisible by 5. The rule for divisibility by 7 is tricky. In short, remove the last digit double it and subtract that from the number remaining; repeat until a number is reached that can identified as divisible or not divisible by 7. For 159,137; 15,913 – 2(7) = 15,899; then 1589 – 2(9) = 1571; then 157 – 2(1) = 155; and, finally, 15 – 2(5) = 5, which is not divisible by 7, so 159,137 is not divisible by either. I determine of a number is divisible by 11, alternately add and subtract the digits. For this number, 1 – 5 + 9 – 1 + 3 – 7 = 0, which is a multiple of 11, so the number is divisible by 11. In fact, 159,137 = 11 14,467. Then, a calculator can be used to show that 14,467 = 17 23 37. The difference between the two greatest factors is 37 – 23 = 14. 9. Last Prime Date. 2/2 and 11/29/ The smallest prime is 2, so 2/2 would be the first prime date. The largest prime less than or equal to 12 is 11. November, the eleventh month, has 30 days. The largest prime less than or equal to 30 us 29. 10. Generate fractions by subtracting 1, 2, 3, …, from the numerator and denominator of given fraction 19/24. Stop when the result is a fraction equivalent to ¾. Subtracting 1 gives 18/23. Subtracting 2 gives 17/22, subtracting 3 gives 16/21, subtracting 4 gives 15/20 = 3/4. The number 4 was subtracted from the numerator and denominator. 11. There are 4 I’s in the eleven-letter word MISSISSIPPI, so 4/11 of the word’s letters are I’s. 12. If x is 5, we have 5/5 = 1 before adding 3’s to numerator and denominator and 8/8 = 1 afterward. To find the algebraically, we could set up the equation x = x + 3 and solve for x. 5 5+ 3 3 13. In the simplified fraction , the numerator is 2 less than the denominator. In the equivalent fraction, 5 the numerator 60 less than the denominator. This means that 2 units in the simplified fraction equal 60 units in the equivalent fraction, so 3 units would equal 90. The equivalent fraction must be 90 , 150 and the numerator is 90. 14. Sample answers: 48xy2z and 72x2y3, 24x5y3z and 96xy2. 15. Let’s assume we do not have any idea where to start. We know we need a fraction equivalent to 3 . 7 6 Multiplying the numerator and denominator by 2 gives us the equivalent fraction , but the sum of 14 the numerator and denominator is only 20. The next fraction is 9 , but again its sum of 30 falls way 21 12 short of 160. The next fraction is , and gives a sum of 40. We are still far from the sum of 160 we 28 need, but notice that our sum is increasing by 10 each time we increase the factor by which we are multiplying the numerator and denominator. We just multiplied both by 4. We need the sum to go up another 160 – 40 = 120, which is 12 more 10s. Raising our factor of 4 by 12, we get 16. Let’s try multiplying the numerator and denominator by 16. We get 48 , which has the correct sum of 160. 112 Rather than using the above Guess, Check & Revise method, we could set up the situation algebraically. We know we have to multiply the numerator and denominator each by the same number, x, in order to get an equivalent fraction. We also know the hew numerator and denominator must add to 160, so 3x + 7x = 160 or 10x = 160 or x = 16. We have determined we must multiply the numerator and denominator by 16, and this yields 48 . Can we see now why the sums in our first solution kept 112 increasing by 10? 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 6 The Math Forum @ Drexel (http://mathforum.org/) 1 – 5, 8 , 10 – 13 , 15 Math Counts (http://mathcounts.org) 7 Friedland, Aaron J. Puzzles In Logic & Math. New York City: Dover Publication, 1970. 9 Collier, C. Patrick. Menu Collection Problems Adapted from Mathematics Teaching in the Middle School. New York: National Council of Teachers of Mathematics, 2000. Print. 14 Larson, Boswell, Kanold, and Stiff. Mathematics Concepts and Skills Course 2 Math Log. McDougal Littell, 2001.