Solve a problem using concepts of prime and composite numbers.

Read the problem solving strategy guide and then from "Prime and Composite Numbers" solve problems #10, 17, 19, 20, 24, and from "Greatest Common Divisor and Least Common Multiple" solve 2,3,4, 8 and 11.

Assessment 4-2B 1. Find the least whole number greater than 0 that is divisible 9. a. Use the Fundamental theorem of Arithmetic to prove tha by four different primes. if 4 n and 9|n, then 36|n. * 2. Determine which of the following numbers are primes: b. When is it true that if a | n and b| n, then ab | n? Justify a. 89 Prime b. 147 Not prime your answer. * c. 159 Not prime d. 187 Not prime 10. Find the greatest four-digit whole number that has exactly e. 2 . 3 . 5 . 7 + 5 * f. 2 . 3 . 5 .7 - 5 Not prime three positive factors. 972 = 9409 3. Use a factor tree to find the prime factorization for each of 11. Show that if I were considered a prime, every number would the following: have more than one prime factorization. * a. 304 * b. 1570 * c. 9550 * 12. Is it possible to find non-zero whole numbers x, y, and z such 4. a. Fill in the missing numbers in the following factor tree: that 2" - 37 = 52? Why or why not? * 13. Show that there are infinitely many composite numbers in 0 the arithmetic sequence 1, 5, 9, 13, 17, . . . . * 14. If 2N = 26 . 35 - 54. 73. 11 , explain why 2 . 3 . 5 . 7 . 11 is a factor of N. * 15. Is 32 . 24 a factor of 33 . 22? Explain why or why not. * 16. Explain why each of the following numbers is composite: a. 7 . 11 . 13 . 17 + 17 * b. 10! + k, where k = 2, 3, 4, 5, 6, 7, 8, 9, or 10 (10! is the product of the whole numbers 1 through 10.) * 17. Explain why 2" . 5' . 9 is not a prime factorization and find the prime factorization of the number. * 18. A prime such as 7331 is a superprime because any integers b. How could you find the top number without finding the obtained by deleting digits from the right of 7331 are prime; other two numbers? * namely, 733, 73, and 7. 5. What is the greatest prime you must consider to test whether a. For a prime to be a superprime, what digits cannot appea 503 is prime? 19 in the number? 4, 6, 8, 0 6. Find the prime factorizations of the following: b. Of the digits that can appear in a superprime, what digit a. 1001 7 . 11 . 13 b. 1001- 72- 112 . 132 c. 99910 330. 3710 cannot be the leftmost digit of a superprime? 1 and 9 d. 11110 - 1119 2 - 39- 5 - 11-379 c. Find all of the two-digit superprimes. * 7. Suppose the 435 members of the House of Representatives d. Find a three-digit superprime other than 733. * are placed on committees consisting of more than 2 members 19. Gina wants to plant fruit trees in a rectangular array. For each of but fewer than 30 members. Each committee is to have an the following numbers of trees, find all possible numbers of rows equal number of members and each member is to be on only if each row is to have the same number of trees: one committee. a. 15 * b. 20 * c. 19 * d. 100 * a. What size committees are possible? 3, 5, 15, 29 people 20. Find the prime factorizations of each of the following: b. How many committees are there of each size? * a. 164- 814. 6 222. 322 b. 8- 325 237 8. Find the least natural number divisible by each natural c. 22 . 35. 755 + 24. 34. 755 22. 34.756 number less than or equal to 12. 27,720 Mathematical Connections 4-2 Communication 1. Explain why the product of any three consecutive whole A woman with a basket of eggs finds that if she removes the numbers is divisible by 6. * eggs from the basket 3 or 5 at a time, there is always 1 egg 2. Explain why the product of any four consecutive whole left. However, if she removes the eggs 7 at a time, there are numbers is divisible by 24. * no eggs left. If the basket holds up to 100 eggs, how many 3. In order to test for divisibility by 12, one student checked eggs does she have? Explain your reasoning. to determine divisibility by 3 and 4; another checked for 7. Explain why, when a number is composite, its least whole divisibility by 2 and 6. Are both students using a correct number divisor, other than 1, must be prime. * approach to divisibility by 12? Why or why not? * 3. Euclid proved that given any finite list of primes, there exists 4. In the Sieve of Eratosthenes for numbers less than 100, a prime not in the list. Read the following argument and explain why, after crossing out all the multiples of 2, 3, 5, and answer the questions that follow. 7, the remaining numbers are primes. Let 2, 3, 5, 7, ..., p be a list of all the primes less than 5. Let M = 2 . 3 . 5 .7 + 11 . 13 . 17 . 19. Without multiplying, or equal to a certain prime p. We will show that there exists a show that none of the primes less than or equal to 19 divides M. *