Competition problem — Number theory

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Number theory Difficulty 3.9 AMC 10/12 Find the answer China

Let $a$ , $b$ , $c$ , $d$ be positive integers and $\log_a b = \frac{3}{2}$ , $\log_c d = \frac{5}{4}$ . If $a-c=9$ , then $b-d=$ ________.

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

We have $b = a^{\frac{3}{2}}$ , $d = c^{\frac{5}{4}}$ from the assumption. We may assume that $a = x^2$ , $c = y^4$ with $x$ and $y$ being positive integers, since $a$ , $b$ , $c$ , $d$ are all positive integers. Then $a-c = x^2 - y^4 = (x-y^2)(x+y^2) = 9$ . It follows that $(x-y^2, x+y^2) = (1, 9)$ . So we obtain the solution $x=5$ , $y=2$ . That is, $b-d = x^3 - y^5 = 125-32=93$ .

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