7 6 n − 6 6 n , where n is an integer > 0 , is divisible by (CAT 2002) Options: A . &nbsp 13 B . &nbsp 127 C . &nbsp 559 D . &nbsp All of these E . &nbsp 81

76n−66n, where n is an integer >0, is divisible by (CAT 2002)

Question

$7^{6 n} - 6^{6 n},$ where n is an integer $> 0$ , is divisible by (CAT 2002)

Answer: Option D
:
D
Ans: (d)

For $n = 1, 7^{6} - 6^{6} = (7^{3})^{2} - (6^{3})^{2};$

$= (7^{3} - 6^{3}) (7^{3} + 6^{3}) = (343 - 216) (343 + 216); = 127 * 559 = 127 * 13 * 43.$ Hence it is divisible by $127, 13, 559.$

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