The number of ways of selecting two numbers a and b, a ∈ {2, 4, 6, ......, 100} and b ∈ {1, 3, 5, ......, 99} such that 2 is the remainder when a + b is divided by 23 is:

Let a be from the set of even numbers up to 100 and b be from the set of odd numbers up to 99. We need a + b = 23k + 2, where a + b is odd, so k must be odd, giving possible sums of 25, 71, and 117. For the sum 25, there are 12 valid pairs; for 71, there are 36 valid pairs; and for 117, there are 60 valid pairs. Adding these gives 12 + 36 + 60 = 108 total ways. The result is 108.

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