Given any seven integers there will be four for which the su

Given any seven integers, there will be four for which the sum of the squares of those integers is divisible by 4.

Solution

Use Pigeonhole Principle here. Any number when divided by 4 can have 4 possible remainder {0,1,2,3}. Here 7 numbers are given, hence some 3 of them must have same remainder.
Also, in seven squares we can get an addition square of 4k,4k+1, 4k+2, 4k+3 which when combined with 3 squares will be divisible by 4.