Donnie Smith Mohacsy MAT 243 ONLINE A Fall 2017 Assignment Unit13 Prime Numbers GCD Euklid Algorithm LCM due 09/17/2017 at 12:00am MST 4. (1 point) The value of the Euler function ( is the Greek letter phi) at the positive integer n is defined to be the number of positive integers less than or equal to n that are relatively prime to n. For example fon n=14, we have {1, 3, 5, 9, 11, 13} are the positive integers less than or equal to 14 which are relatively prime to 14. Thus (14) = 6. Find: (5) (25) (15) (45) 1. (1 point) Are the following integers primes? Enter \"1\" for a prime and \"0\" otherwise. 1. 2. 3. 4. 5. 6. 1 164 133 49 187 169 5. (1 point) Suppose an integer x is divisible by 21. What other positive integers must also divide x? [Hint: Think about the transitivity of \"divides.\" Enter your answer as a comma-separated list.] 2. (1 point) Find the prime factorization of the follwing numbers: (write p0 if a prime does not appear in the given number.) 300 = 2a 3b 5c 7d 11e 13 f 17g 19h where b= a= c= d= e= f= g= Suppose an integer a is divisible by 16. What other positive integers must also divide a? [Enter your answer as a comma-separated list.] h= 3920 = 2a 3b 5c 7d 11e 13 f 17g 19h where a= b= c= d= e= f= g= 6. (1 point) Assuming that p, q, r are distinct primes, how many positive divisors does m have in the following cases: If m = q3 h= If m = p2 q2 374 = 2a 3b 5c 7d 11e 13 f 17g 19h where a= b= c= d= e= f= g= If m = pqr h= [Hint: Think of a particular number with the given factorization.] 347633 = 2a 3b 5c 7d 11e 13 f 17g 19h where a= b= c= d= e= f= g= 7. (1 point) Suppose an integer has the factorization p2 q, where p and q are unique primes. How many positive divisors does this integer have? h= What is the smallest nonnegative integer with this factorization? 3. (1 point) What are the greatest common divisors of the following pairs of integers? (a) 25 32 55 and 2 34 55 Answer = (b) 26 3 5 7 11 13 and 26 58 77 17 Answer = (c) 25 7 and 52 13 Answer = 8. (1 point) What is the smallest integer, larger than 1, which is both a perfect square, and a perfect cube. answer: 1 What is the smallest integer with prime factorization pq, where p and q are distinct prime numbers? answer: The prime factors of 21 are The prime factors of 47 are The prime factors of 56 are The prime factors of 52 are What are the next three smallest integers with prime factorization pq with p and q distinct? , , Largest Smallest The prime factors of 35 are The prime factors of 65 are The prime factors of 59 are What is the smallest integer with prime factorization pqr, where p, q, and r are distinct prime numbers? answer: The prime factors of 53 are The prime factors of 54 are 12. (1 point) Let a = p2 q and b = p q r where p, q, r are distinct primes. For the following questions, indicate the correct exponent. Enter 0 if necessary. Determine gcd(a,b) pn for n = qn for n = rn for n = Determine lcm(a,b) pn for n = qn for n = rn for n = 9. (1 point) For n = 1470, find the prime factorization. [Note: Enter your answer as a comma-separated list. If the factorization is 22 32 , enter your answer as 2, 2, 3, 3] 10. (1 point) What is the prime factorization of m? m 7 15 17 25 30 = 14 21 34 75 85 95 Find the prime factorizations of s and t for the smallest integer solutions of 60 s = 126 t s= t= 13. (1 point) Determine the greatest common divisor of 357 and 221. gcd(357, 221) = Note: Enter your answers as a comma-separated list. 2 2 3 should be entered as \"2,2,3\" 14. (1 point) Determine the least common multiple of 17 and 32. 11. (1 point) For each of the following numbers write down the number's prime factors as a comma-separated list (so, \"5\" or \"2,3\" for 25 and 12 respectively, but without the quotes). lcm(17, 32) = c Generated by WeBWorK, http://webwork.maa.org, Mathematical Association of America 2