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1. 7⁶ⁿ- 6⁶ⁿ, where n is an integer >0, is divisible by

A. 13

B. 127

C. 559

D. All of these

Solution:

\begin{aligned} 7^{6 n} - 6^{6 n} \\ = 7^{6} - 6^{6} \\ = {(7^{3})}^{2} - {(6^{3})}^{2} \\ = (7^{3} - 6^{3}) (7^{3} + 6^{3}) \\ = (343 - 216) \times (343 + 216) \\ = 127 \times 559 \\ = 127 \times 13 \times 43 \end{aligned}

Clearly, it is divisible by 127, 13 as well as 559

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