Subject: General Aptitude · Chapter: Number System · Exam: · Marks: · Difficulty:
$7^{6 n}-6^{6 n}$, where $n$ is an integer > 0, is divisible by
(a)13
(b)127
(c)559
(d)All of these
Answer
Answer (as printed): D
Explanation
When $n$ is even, $\left(x^{n}-a^{n}\right)$ is divisible by both $(x-a)$ as well as $(x+a)$. Now, $\left(7^{6 n}-6^{6 n}\right)=\left[\left(7^{3}\right)^{2 n}-\left(6^{3}\right)^{2 n}\right]=\left[(343)^{2 n}-(216)^{2 n}\right]$. $\therefore\left(7^{6 n}-6^{6 n}\right)$ is divisible by both $(7-6)$ and $(7+6)$ $\Rightarrow\left(7^{6 n}-6^{6 n}\right)$ is divisible by 13. And, $\left[(343)^{2 n}-(216)^{2 n}\right]$ is divisible by both $(343-216)$ and (343 + 216) $\Rightarrow\left(7^{6 n}-6^{6 n}\right)$ is divisible by both 127 and 559.
Explanation as extracted from the printed page; notation may be imperfect.