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Let m > 1 be a positive integer.  Find the remainder when 2^m + 5^(m - 1) + 7^(m - 2) is divided by 3.

Dec 18, 2019

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Let $$m > 1$$ be a positive integer.

Find the remainder when $$2^m + 5^{m - 1} + 7^{m - 2}$$ is divided by 3.

$$\begin{array}{|rcll|} \hline && 2^m + 5^{m - 1} + 7^{m - 2} \\ &\equiv& (3-1)^m + (6-1)^{m - 1} + (6+1)^{m - 2} \\ &\equiv& \binom{m}{0}3^{m} - \binom{m}{1}3^{m-1}+ \binom{m}{2}3^{m-2}+-\ldots \pm \binom{m}{m} \\ && +\binom{m-1}{0}6^{m-1} - \binom{m-1}{1}6^{m-2}+ \binom{m-1}{2}6^{m-3}+-\ldots \mp \binom{m-1}{m-1} \\ && +\binom{m-2}{0}6^{m-2} \binom{m-2}{1}6^{m-3}+ \binom{m-2}{2}6^{m-4}+\ldots + \binom{m-2}{m-2} \\ &\equiv& \underbrace{\pm \binom{m}{m} \mp \binom{m-1}{m-1}}_{=0}+ \underbrace{\binom{m-2}{m-2}}_{=1} \\ &\equiv& \mathbf{1} \pmod{ 3 } \\ \hline \end{array}$$

Dec 18, 2019