Question

What is the remainder when $2^{125}$ is divided by 11?

• A. 7
• B. 10
• C. 8
• D. 6

Answer 1

Answer:

B. 10

$=\frac{2^{125}}{11}$

$=\frac{(2^5)^{25}}{11}$

$=\frac{32^{25}}{11}$

Remainder when ‘32’ is divided by ‘11’ = 10

Now,

$=\frac{(32)^{25}}{11}$

$=\frac{(10^{24}\times{10})}{11}$

$=(\frac{(10^{24}}{11})\times(\frac{10}{11})$

$=\frac{(10^2)^{12}}{11}\times\frac{10}{11}$

$=\frac{(10^2)^12}{11}\times\frac{10}{11}$

$=\frac{(100^{12}}{11}\times\frac{10}{11}$

Remainder when ‘100’ is divided by ‘11’ = 1

Now,

$=\frac{1^{12}}{11}\times{10}{11}$

$\frac{1}{11}\times\frac{10}{11}$

$=\frac{(1\times{10})}{11}$

Hence, required remainder = 10

Alternate method:

Using Euler's Method,

To find the remainder when 2¹²⁵ by 11, we have to find the Euler's number of 11.

We know that the Euler number of any prime number (n) is (n - 1).

Euler's number of 11 is 10.

When we divide 125 by 10 then we got remainder 5.

2⁵ = 32

Now divide 32 by 11 and find the remainder.

When we divide 32 by 11 then the remainder is 10.