Let $a$, $b$ be positive integers. Suppose that $x$, $y$ are the smallest positive integers such that $ax - by = 0$. Prove that $x = \frac{b}{gcd(a, b)}$ and $y = \frac{a}{gcd(a, b)}$
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This means that x is a multiple of gcd(a, b) plus some additional positive integer, let's call it k. So we can write x = gcd(a, b) + k. Now let's substitute this value of x into the equation ax - by = 0: a(gcd(a, b) + k) - by = 0 Expanding this equation, we Show more…
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