If $ 53^{53}-33^{3} $ is divided by $ 10 $, then the remainder obtained is

Question

If $ 53^{53}-33^{3} $ is divided by $ 10 $, then the remainder obtained is,

MARKS App · Accepted Answer

We know that, x n - y n = x - y x n - 1 + x n - 2 y + . . . . + x y n - 2 + y n - 1 i.e. x n - y n is divisible by x - y 5 3 53 - 3 3 3 = 5 3 53 - 3 53 + 3 53 - 3 3 3 + 3 3 - 3 3 = 5 3 53 - 3 53 + 3 53 - 3 3 - 3 3 3 - 3 3 Clearly, 1 s t bracket is divisible by 50 and the last one is divisible by 30 . So, both are divisible by 10 . 3 53 - 3 3 = 3 3 ⋅ 3 50 - 3 3 = 3 3 3 50 - 1 = 3 3 10 - 1 25 - 1 = 27 10 - 1 25 - 27 = 27 10 k - 1 - 27 = 270 k - 54 = 270 k - 60 + 6 = 10 μ + 6 . Hence, the remainder obtained is 6 .

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