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Question

Let Sm denote the sum of the squares of the first m natural numbers. For how many values of m < 100, is Sm a multiple of 4?

A
50
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B
25
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C
36
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D
24
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Solution

The correct option is C 24
We need the expression (n(n+1)(2n+1))6 to be a multiple of 4. For this to occur, the numerator of the above expression should be a multiple of 8. In the expression n (n + 1) (2n + 1), 2n + 1 would always be an odd number. Also, amongst n and (n + 1), one number would be odd and the other would be even. Since, we need n (n + 1) (2n + 1) to be a multiple of 8, we would need either n or (n + 1) to be a multiple of 8(while at the same time it should be below 100). Thus, we get the number series n = 7, 8, 15, 16, 23, 24, .... 95, 96. Since there are 12 multiples of 8 below 100, the required answer is 12×2=24

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