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Look at it this way, sum of squares of first n no = [n(n+1)(2n+1)]/6. You know this, right? This is represented in the question to be K.
So now, K to be divisible by 4, must have atleast one out of n, n+1 to be divisbile by 4*2 (because it should also be divisible by 2*3) or 8. Dont worry about divisbility by 3, as one of the three numbers, n, n+1 and 2n+1 is always divisible by 3.
So we get pairs such as m =7 or 8, 15 or 16 ... upto 47 or 48. Thus total number of values of m satisfying condition for K's divisibility by 4, will be 12.