sol4 - MATH 235 Algebra 1 Solutions to Assignment 4...

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MATH 235 – Algebra 1 Solutions to Assignment 4 October 10, 2007 Solution of 1. Let a, b be two positive integers and write them as a = k Y i =1 p r i i and b = k Y i =1 p s i i where the p i ’s are distinct primes and r i , s i 0 for i = 1 , . . . , k . (Note that it is indeed possible to take the same primes in both factorizations because we’re allowing the exponents to be 0.) (1) From Proposition 10.6, we know that the set of common divisors of a and b is ± k Y i =1 p u i i ² ² ² ² u i 0 , u i r i , u i s i ³ . The GCD being by definition the maximal element in this set, it follows that the GCD of a and b is the common divisor of a and b having the maximal possible exponents u i in its prime factorization, that is, ( a, b ) = k Y i =1 p min( r i ,s i ) i . (2) Again using Proposition 10.6, we know that the set of common multiples of a and b is ± k Y i =1 p u i i ² ² ² ² u i r i , u i s i ³ . The LCM being by definition the minimal element in this set, it follows that the LCM of a and b is the common multiple of a and b having the minimal possible exponents u i in its prime factorization, that is, [ a, b ] = k Y i =1 p max( r i ,s i ) i . 1
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Solution of 2. (1) The fact that [ a, b ] divides any common multiple of a and b follows readily from the description given in the solution of question 1, but we give here another proof. Let k be an integer such that a | k and b | k . Applying the division algorithm, we can write k = q [ a, b ] + r with 0 r < [ a, b ] . But from
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This note was uploaded on 04/18/2008 for the course MATH 235 taught by Professor Goren during the Fall '07 term at McGill.

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sol4 - MATH 235 Algebra 1 Solutions to Assignment 4...

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