Question

\( 10^{68} \) is divided by 13, the remainder is  is

• A. 9
• B. 4
• C. 5
• D. 8

Answer 1

The Correct Option is A

To solve this, we need to find the remainder when $10^{68}$ is divided by 13. This can be done using modular arithmetic. We first calculate the powers of 10 modulo 13:
\(10^1  mod 13 = 10 \\ 10^2  mod  13 = 100  mod  13 = 9 \\ 10^3  mod  13 = 1000  mod  13 = 12 \\ 10^4  mod  13 = 10000  mod  13 = 3 \\ 10^5  mod  13 = 100000  mod  13 = 4 \\ 10^6  mod  13 = 1000000  mod  13 = 1 \)
Since $10^6 \equiv 1 \pmod{13}$, we can simplify $10^{68} \pmod{13}$ by noticing that $68 = 6 \times 11 + 2$. Therefore:
\[ 10^{68} = 10^{6 \times 11 + 2} = (10^6)^{11} \times 10^2 \]
Using $10^6 \equiv 1 \pmod{13}$, this simplifies to:
\[ 10^{68} \equiv 1^{11} \times 10^2 \equiv 10^2 \equiv 9 \pmod{13} \]
Thus, the remainder when $10^{68}$ is divided by 13 is 9.