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[x] and function gcd

Source: 7-th Taiwanese Mathematical Olympiad 1998

January 19, 2007
functionnumber theorygreatest common divisornumber theory proposed

Problem Statement

Let m,nm,n are positive integers. a)Prove that (m,n)=2∑k=0m−1[knm]+m+n−mn(m,n)=2\sum_{k=0}^{m-1}[\frac{kn}{m}]+m+n-mn. b)If m,n≥2m,n\geq 2, prove that ∑k=0m−1[knm]=∑k=0n−1[kmn]\sum_{k=0}^{m-1}[\frac{kn}{m}]=\sum_{k=0}^{n-1}[\frac{km}{n}].
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