Subject: General Aptitude · Chapter: Number System · Exam: · Marks: · Difficulty:
What is the remainder when $4^{61}$ is divided by 51?
(a)20
(b)41
(c)50
(d)None of these
Answer
Answer (as printed): D
Explanation
$4^{61}=4 \times 4^{60}=4 \times\left(4^{4}\right)^{15}=4 \times(256)^{15}$. Now, $\left(x^{n}-a^{n}\right)$ is divisible by $(x-a)$ for all values of $n$. $\therefore\left(256^{15}-1\right)$ is divisible by $(256-1)$ i.e. 255 and hence by 51. ⇒ On dividing (256) ${ }^{15}$ by 51, we get 1 as remainder ⇒ On dividing $4{ }^{60}$ by 51, we get 1 as remainder ⇒ On dividing $4{ }^{61}$ by 51, remainder obtained $=(4 \times 1)=4$.
Explanation as extracted from the printed page; notation may be imperfect.