Number System Question 2 - CAT 2024

Step-by-Step Explanation

To find the remainder of a high power divided by 13, look for a small power of the base close to a multiple of 13.

Notice that:

10³ = 1000

Dividing 1000 by 13 gives 13 × 77 = 1001, which leaves a remainder of - 1:

10³ ≡ - 1 (mod 13)

Now, break the exponent 68 down into multiples of 3:

68 = 3 × 22 + 2

10⁶⁸ = ( 10³ )²² × 10²

Apply the modulo operator to the components:

10⁶⁸ ≡ ( - 1)²² × 100 (mod 13)

10⁶⁸ ≡ 1 × 100 = 100 (mod 13)

Finally, find the remainder of 100 divided by 13:

100 = 13 × 7 + 9 ⇒ Remainder = 9

Correct Option: C