Suppose we want to pick two numbers from {1,2,⋯,100} randomly. What is the probability that the sum of the two picked numbers is divisible by 4?

2 numbers out of 100 can be summed up in:

100 C 2 ==4,950 ways

Out of 4,950 ways, there are 1,225 ways of adding up 2 numbers whose sum is divisible by 4

So, the probability is: 1,225 / 4,950 ==25 / 99