Question

Find the remainder when $2^{245}$ is divided by 11.

• A. 11
• B. 2
• C. 10
• D. 1

Answer 1

The correct option is C 10

Since the number 2 is not divisible by the prime 11, so we get
$2^{10}$ $\equiv$ 1 (mod 11) (1)
by Fermat’s Little Theorem. By the division algorithm,
245 = 24 $\times$ 10 + 5.
Since $2^{245}=2^{(24\times 10+5)}=(2^{10}{)}^{24}\times 2^5$, then
$2^{245}=1^{24}\times 2^5\equiv 1\times 32\equiv$ 10 (mod 11). [using (1)]
Thus $2^{245}$ (mod 11) = 10. Hence option c) is the answer.