Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 5.1 AIME, harder Find the answer

For integers $a, b, c, d$ , let $f(a, b, c, d)$ denote the number of ordered pairs of integers $(x, y) \in \{1,2,3,4,5\}^{2}$ such that $a x+b y$ and $c x+d y$ are both divisible by 5. Find the sum of all possible values of $f(a, b, c, d)$ .

Your answer

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

Solution

Standard linear algebra over the field $\mathbb{F}_{5}$ (the integers modulo 5). The dimension of the solution set is at least 0 and at most 2, and any intermediate value can also be attained. So the answer is $1+5+5^{2}=31$ . This also can be easily reformulated in more concrete equation/congruence-solving terms, especially since there are few variables/equations.

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