M135W09A5

M135W09A5 - c then ab | c 6 A positive integer n is called...

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MATH 135 Winter 2009 Assignment #5 Due: Wednesday 25 February 2009, 8:20 a.m. Hand-In Problems 1. In each part, state the answer. No justifcation is necessary. (a) Determine all non-negative solutions to the linear Diophantine equation 133 x + 315 y = 98000. (b) Determine all non-negative solutions to the linear Diophantine equation 133 x + 315 y = 98000 with the additional property that x 640 - 2 y . 2. Suppose that f ( x ) is a polynomial o± degree n with real coe²cients and suppose that g ( x ) = x - c ±or some c R . (a) When f ( x ) is divided by g ( x ) we obtain a quotient q ( x ) and a remainder r ( x ). Explain why, in this case, r ( x ) is a constant r ( x ) = r R . (b) Prove that r = f ( c ). 3. (a) Find gcd(10! , 3 10 ). (b) Find gcd(3 100 , 100 3 ). 4. (a) Determine the prime ±actorizations o± 39 , 204 , 000 and 9 , 922 , 500. (b) Determine the number o± distinct prime divisors o± 9 , 922 , 500. (c) Determine the number o± distinct positive common divisors o± 39 , 204 , 000 and 9 , 922 , 500 that are multiples o± 10. 5. Let a, b, c P with gcd( a, b ) = 1 . Prove that i± a | c and b |

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Unformatted text preview: c , then ab | c . 6. A positive integer n is called a per±ect square i± n = m 2 ±or some integer m . (a) Prove that the positive integer n is a per±ect square i± and only i± when we ±actor it into a product o± powers o± distinct primes n = p k 1 1 p k 2 2 · · · p k ℓ ℓ all o± the k i are even. (b) Suppose that a, b, c ∈ P , with gcd( a, b ) = 1 and ab = c 2 . Prove that a and b are per±ect squares. 7. (a) What is the remainder when 8 20 is divided by 3? (b) Is 6 17 + 17 6 divisible by 3 or 7? 8. Which o± the ±ollowing integers are congruent modulo 6?-147 ,-91 ,-22 ,-14 , 2 , 4 , 5 , 21 , 185. Recommended Problems 1. Text, page 50 #42. 2. Text, page 51 #44. 3. Text, page 52 #75. 4. Text, page 52 #79. 5. Text, page 54 #92. 6. Prove that if a | b , then a 2 | b 2 . 7. Prove that if a 2 | b 2 , then a | b . 8. Text page 84 #56. 9. Text page 84 #57....
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This note was uploaded on 02/05/2009 for the course MATH 101 taught by Professor Johnson during the Spring '08 term at Adelphi.

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M135W09A5 - c then ab | c 6 A positive integer n is called...

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