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Find the last digit of 9¹²³⁴³³⁴⁴³³³⁴⁴⁴³³³³⁴⁴⁴⁴²¹

1 9 3 0

Rkbm Rizmi by Rkbm Rizmi , 23, Bangladesh

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2 solutions

Arian Tashakkor
May 4, 2015

9 n = ( 10 − 1 ) n → n=2k+1 → 9 n ≡ ( 10 − 1 ) n ≡ 10 − 1 ≡ 9 ( m o d 10 ) 9^n = (10-1)^n \rightarrow \text {n=2k+1} \rightarrow 9^n \equiv (10-1)^n \equiv 10-1 \equiv 9 \pmod{10}

Not quite sure if this was the solution you had in mind or not but anyways ...

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Abyoso Hapsoro
May 4, 2015

9 a { 9 }^{ a } will yield in 9 for odd values of a a and 1 for even values of a a . Therefore, since 123433443334443333444421 is odd, the answer is 9

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Moderator note:

There's a slightly simpler approach. The answer will be immediate once you know the basics of modular arithmetic. Hint: 9 = 10 − 1 9 = 10 - 1 .

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