What is the remainder when $(1^{1} + 2^{2} + 3^{3} + . . . + 100^{100})$ is divided by 4?

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Solution

There are 50 odd numbers and 50 even numbers. Every even number will be divisible by 4. Further, in odd numbers, half of them have 1 as the remainder and half of them have - 1 as the remainder. Then, the overall remainder is zero.

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The remainder obtained when $(1^{1} + 2^{2} + 3^{3} + 4^{4} + 5^{5} + \dots 10^{10})$ is divided by 4 is

32^{33^{34}}

11^{10} - 11

99 \times 11

2

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