What is the remainder when the following sum is divided by 4?

The sum is:

Sol: The beauty of the solution lies in application of Modular Arithmetic. it is a simple question if the student has some good knack in Modular Arithmetic.

The equivalence class mod 4 has four integer members 0, 1, 2, 3.

The member 0 will belong to set {4, 8, 12, ….,96, 100}, as all these number when divided by 4 will give remainder equal to 0.

Hence, all these numbers {4, 8, 12, ….,96, 100}, when raised to power 5 and divided by 4 gives remainder equal to 0, as 0 raised to any power is 0. ….. (1)

The member 1 will belong to set {1, 5, 9, …., 97}, as all these number when divided by 4 will give remainder equal to 1.

Hence, all these numbers {1, 5, 9, …., 97}, when raised to power 5 and divided by 4 will give remainder equal to 1 as 1 raised to any power is 1. ….. (2)

The member 2 will belong to set {2, 6, 10, …., 98}, i.e. (4n-2) as all these number when divided by 4 will give remainder equal to 2.

Hence, all these numbers {2, 6, 10, …., 98}, when raised to power 5 and divided by 4 will give remainder equal to 0. ….. (3)

The member 3 will belong to set {3, 7, 11, …., 99}, i.e. (4n-1) as all these number when divided by 4 will give remainder equal to 3.

Hence, all these numbers {3, 7, 11, …., 99}, when raised to power 5 and divided by 4 will give remainder equal to 3. ….. (4)

From (1), (2), (3) and (4), total of remainders = 0 + 1 + 0 + 3 = 4

Hence required remainder = 0