Let x,y be distinct integers with 1 x25, 1 y25.

Question

Let $x,y$ be distinct integers with $1\le x\le25$, $1\le y\le25$. The number of ways of choosing $x,y$ such that $x+y$ is divisible by 5 is \_\_\_\_.

Answer: 120

Accepted Answer

Each residue class mod 5 (1--5, 6--10, ..., 21--25) contains 5 numbers. $x+y$ divisible by 5 requires residues summing to 0 mod 5: pairs within the same class (residue 0 with 0) or complementary classes (1\&4, 2\&3).

Counting ordered pairs with $x
e y$: for residue-0 class: $5	imes4=20$; for each complementary pair of classes: $5	imes5	imes2=50$ (two such complementary pairs: 1--4, 2--3), giving $50	imes2=100$.

Total $=20+100=120$.

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