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Choosing two numbers whose sum leaves a given remainder

This type of question, asked 2 times: 25 Jan 2023, 30 Jan 2023. All 2 questions below, oldest first.

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  1. Question 1

    25 Jan 2023

    87
    Let xx and yy be distinct integers where 1≤x≤251 \leq x \leq 25 and 1≤y≤251 \leq y \leq 25. Then, the number of ways of choosing xx and yy such that x+yx+y is divisible by 5, is

  2. Question 2

    30 Jan 2023

    162
    The number of ways of selecting two numbers aa and b,a∈{2,4,6,…,100}\mathbf{b,a \in \{ 2,4,6,\ldots,100\}} and b∈{1,3,5,…,99}\mathbf{b \in \{ 1,3,5,\ldots,99\}} such that 2 is the remainder when a+ba + b is divided by 23 is

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