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Question Number 19499 by tawa tawa last updated on 12/Aug/17

Find the last digit of  2^(253)

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{last}\:\mathrm{digit}\:\mathrm{of}\:\:\mathrm{2}^{\mathrm{253}} \\ $$

Answered by Tinkutara last updated on 12/Aug/17

Cyclicity of 2 = 4 (2, 4, 8, 6)  253 ≡ 1 (mod 4)  ∴ 2^(253)  ≡ 2^1  ≡ 2  Last digit = 2

$$\mathrm{Cyclicity}\:\mathrm{of}\:\mathrm{2}\:=\:\mathrm{4}\:\left(\mathrm{2},\:\mathrm{4},\:\mathrm{8},\:\mathrm{6}\right) \\ $$$$\mathrm{253}\:\equiv\:\mathrm{1}\:\left(\mathrm{mod}\:\mathrm{4}\right) \\ $$$$\therefore\:\mathrm{2}^{\mathrm{253}} \:\equiv\:\mathrm{2}^{\mathrm{1}} \:\equiv\:\mathrm{2} \\ $$$$\mathrm{Last}\:\mathrm{digit}\:=\:\mathrm{2} \\ $$

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