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  • Find the rightmost digit of  <img alt="1^n+2^n+3^n+4^n+5^n+6^n+7^n+8^n+9^n(n \text { arbitrary positive integer)}"

Find the rightmost digit of : 1^n+2^n+3^n+4^n+5^n+6^n+7^n+8^n+9^n(n \text { arbitrary positive integer)}

Option: 1

n is not a multiple of 4


Option: 2

n is multiple of 4


Option: 3

n is not multiple of 5


Option: 4

None of these


Answers (1)

best_answer

Modulo 10 easy inductions show that

\begin{aligned} & 1^n+9^n \equiv \begin{cases}0, & \text { if } n \text { is odd } \\ 2, & \text { if } n \text { is even },\end{cases} \\ & 2^n+8^n \equiv \begin{cases}0, & \text { if } n \text { is odd } \\ 8, & \text { if } n \equiv 2(\bmod 4) \\ 2, & \text { if } n \equiv 4(\bmod 4),\end{cases} \\ & 3^n+7^n \equiv \begin{cases}0, & \text { if } n \text { is odd } \\ 8, & \text { if } n \equiv 2(\bmod 4) \\ 2, & \text { if } n \equiv 0(\bmod 4),\end{cases} \\ & 4^n+6^n \equiv \begin{cases}0, & \text { if } n \text { is odd } \\ 2, & \text { if } n \text { is even, and }\end{cases} \\ & 5^n \equiv 5 . \end{aligned}

Thus, 

\sum_{k=1}^9 k^n \equiv \begin{cases}5, & \text { if } n \neq \equiv(\bmod 4) \\ 3, & \text { if } n \equiv 0(\bmod 4)\end{cases}

And as a quick check of the secound case above, 

\sum_{k=1}^9 k^4=1+16+81+256+625+1296+2401+4096+6561=15,333

Noted that the started result is true only when n is not multiple of 4.

Posted by

shivangi.shekhar

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