Question

$10^{68}$ is divided by 13, the remainder is is

• A. 9
• B. 4
• C. 5
• D. 8

Answer 1

Answer: (A)

9

Answer 2

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.