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Unformatted text preview: divisors of m and r. What does this mean if r = 0. Is the theorem still true? Conclude as a corollary: gcd(m,n) = gcd (m,r). Note: As clarified in an email, the integers here refer to the 9
2. Answer: We want to show that any divisor of m and n is also a divisor of m and r and vice versa. Let d be a divisor of m and n. Than means for some integers a and b, m = da and n = db. But we have n = qm +r, so r = n =qm = da – qdb = d(a –qb), so r is divisible by d. And we already assumed that m is divisible by d. In the other direction, assume that d divides both m and r, so m = db and r = dc. Then n = qm + r = qdb+dc = d(qb+c), so n is divisible by d also. QED. Problem 9
4: Use the result of 9
3 for an algorithm to find the gcd of any tw...
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 Winter '11
 JamesKing
 Math, Integers

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