Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 5.2 AIME, harder Prove it United States

Problem:

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 $f(a, b, c, d)$ .

Solution

Solution:

Answer: $31$

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 $1 + 5 + 5^{2} = 31$ .

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