ASSIGNMENT 4 - MATH235, FALL 2007Submit by 16:00, Tuesday, October 9 (use the designated mailbox in Burnside Hall,10thfloor).1. Leta=pr11pr22· · ·prkkandb=ps11ps22· · ·pskk, wherep1, p2, . . . , pkare distinct positive primes and eachri, si≥0. Using unique factorization, prove that(1)(a, b) =pn11pn22· · ·pnkk, whereni= min(ri, si).(2)[a, b] =pt11pt22· · ·ptkk, whereti= max(ri, si).2. The least common multiple of nonzero integersa, bis the smallest positive integermsuch thata|mandb|m. We denote it by lcm(a, b) or [a, b]. Prove that:(1)Ifa|kandb|kthen [a, b]|k.(2)[a, b] =ab(a,b)ifa >0, b >0.3. Prove or disprove: Ifnis an integer andn >2, then there exists a primepsuch thatn < p < n!.4. Find all the primes between 1 and 150. The solution should consist of a list of all the primes + givingthe last prime used to sieve + explanation why you didn’t have to sieve by larger primes.
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