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# ass4 - ASSIGNMENT 4 MATH235 FALL 2007 Submit by 16:00...

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ASSIGNMENT 4 - MATH235, FALL 2007 Submit by 16:00, Tuesday, October 9 (use the designated mailbox in Burnside Hall, 10 th floor). 1. Let a = p r 1 1 p r 2 2 · · · p r k k and b = p s 1 1 p s 2 2 · · · p s k k , where p 1 , p 2 , . . . , p k are distinct positive primes and each r i , s i 0. Using unique factorization, prove that (1) ( a, b ) = p n 1 1 p n 2 2 · · · p n k k , where n i = min( r i , s i ). (2) [ a, b ] = p t 1 1 p t 2 2 · · · p t k k , where t i = max( r i , s i ). 2. The least common multiple of nonzero integers a, b is the smallest positive integer m such that a | m and b | m . We denote it by lcm( a, b ) or [ a, b ]. Prove that: (1) If a | k and b | k then [ a, b ] | k . (2) [ a, b ] = ab ( a,b ) if a > 0 , b > 0. 3. Prove or disprove: If n is an integer and n > 2, then there exists a prime p such that n < p < n !. 4. Find all the primes between 1 and 150. The solution should consist of a list of all the primes + giving the last prime used to sieve + explanation why you didn’t have to sieve by larger primes.
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