Question

If $5^{15}-1$ is divisible by a certain number then which of the following is also divisible by that number

• A. $5^{75}-1$
• B. $5^{70}-1$
• C. $5^{80}-1$
• D. $5^{75}$

Answer 1

Answer: (A)

$5^{75}-1$

Let $5^{15}-1$ be divisible by $N.$

$\Rightarrow 5^{15}-1=\lambda N$

$\Rightarrow 5^{15}=\lambda N+1$

Now, we have to check for all options.

$\Rightarrow ({5^{15})}^5=({\lambda N+1)}^5$

$=\left[\right({\lambda N)}^5+5({\lambda N)}^4+.....+1]$

$\Rightarrow 5^{75}=KN+1$ $;K\in I$

$\Rightarrow (5^{75}-1)=KN$

Hence, $5^{75}-1$ is also divisible by $N.$

Answer is $5^{75}-1.$