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By Tsvetkov V. M.

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Solution. Let a = pai'p22• • 14,and b = gip be the prime factorizations of a and b; since (a,b)= 1, no pi is a qi . If ab = nk , the prime divisors of n are clearly just the pi and qi. ,9k bi = kdi for each i. Thus a = pkic, 1 . 1 34. (a) Let a, b, c be positive integers. Show that if ab, ac, and be are - perfect cubes, then a, b, and c must be perfect cubes. (b) Discuss what happens if we replace "perfect cube" by "perfect kth power. " Solution. (a) We use the Unique Factorization Theorem. For any prime p, let pap be the largest power of p that divides a, and define by and cpanalogously.

Let n = Ijpni' be the prime factorization of n. Prove that n is a perfect - square if and only if each ni is even. Solution. If each ni is even, say, ni = 2ci , then n = ({1K12. Now suppose n is a square, say, n = m2. If m = jI /r', then ni =2m, for each i. 1 33. Prove that if (a, b) = 1 and ab is a kth power, then a and b are each - kth powers. Solution. Let a = pai'p22• • 14,and b = gip be the prime factorizations of a and b; since (a,b)= 1, no pi is a qi . If ab = nk , the prime divisors of n are clearly just the pi and qi.

Since m n is even if m and n are odd, we are dividing by 2 at least every second step, so the algorithm terminates quite rapidly. The Binary GCD Algorithm is particularly efficient on a binary computer. Division is a fairly slow operation, and divisions account for most of the time spent in running the Euclidean Algorithm. On a binary computer, however, division by 2 is fast (simply remove the final 0 in the binary representation of the number). — The Binary GCD Algorithm can be extended in a straightforward way to produce integers x and y such that ax + by = (a,b).

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