Competition problem — Number theory

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Number theory Difficulty 4.5 AIME Find the answer

1. The number of positive integer solutions $(a, b, c)$ for the equation $6 \times\left(5 a^{2}+b^{2}\right)=5 c^{2}$ , satisfying $c \leqslant 20$ , is:

Pick one

Solution

$-1 .(\mathrm{C})$ .
Let $c=6 c_{1}\left(1 \leqslant c_{1} \leqslant 3\right), b=5 b_{1}$ , substituting into the original equation yields $5 b_{1}^{2}+a^{2}=6 c_{1}^{2}$ .
Taking $c_{1}=1,2,3$ respectively, a total of 『 group(s) of solutions are obtained.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.