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 find the remainder when \( 10^{68} \) is divided by 13, we can use Fermat's Little Theorem. The theorem states that if \( p \) is a prime number and \( a \) is an integer not divisible by \( p \), then \( a^{p-1} \equiv 1 \ (\text{mod} \ p) \).

Applying this here:

\( a = 10 \), \( p = 13 \). Therefore, \( 10^{12} \equiv 1 \ (\text{mod} \ 13) \).

We need to express \( 10^{68} \) in terms of powers of 12:

\( 68 \div 12 = 5 \) remainder \( 8 \). So, \( 68 = 5 \times 12 + 8 \).

Therefore, \( 10^{68} = (10^{12})^5 \times 10^8 \equiv 1^5 \times 10^8 \equiv 10^8 \ (\text{mod} \ 13) \).

Now, calculate \( 10^8 \mod 13 \):

\( 10^2 = 100 \equiv 9 \ (\text{mod} \ 13) \)

\( 10^4 = (10^2)^2 \equiv 9^2 = 81 \equiv 3 \ (\text{mod} \ 13) \)

\( 10^8 = (10^4)^2 \equiv 3^2 = 9 \ (\text{mod} \ 13) \)

Hence, the remainder when \( 10^{68} \) is divided by 13 is 9.