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Integers gcd problem

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發問:

Can anyone help me to prove this question. show that if a>0, then (ab,ac)=a(b,c)

最佳解答:

Let b, c be positive integers and (b, c) = m. By definition, m | b and m | c. Therefore, for any positive a > 0, am | ab and am | ac. Hence am ≤ (ab, ac) (Because, by defintion, (ab, ac) is the greatest common divisor of ab and ac). On the other hand, (b, c) = m implies that there are integers x and y such that xb + yc = m. (Because of the Euclid algorithm) Thus, x(ab) + y(ac) = am. By definition, (ab, ac)|ab and (ab, ac)|ac and therefore, (ab, ac) | am. Hence, (ab, ac) ≤ am. Combining the above two paragraphs, we have (ab, ac) = am = a(b, c).

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