Number System Question 3 - CAT 2024

Step-by-Step Explanation

Simplify the base first:

10 ≡ 3 (mod 7)

10¹⁰⁰ ≡ 3¹⁰⁰ (mod 7)

According to Fermat's Little Theorem, if p is a prime number, then aᵖ⁻¹ ≡ 1 (mod p). Since 7 is prime:

3⁷⁻¹ = 3⁶ ≡ 1 (mod 7)

Divide the exponent 100 by the cycle length of 6:

100 = 6 × 16 + 4

3¹⁰⁰ = ( 3⁶ )¹⁶ × 3⁴ ≡ 1¹⁶ × 3⁴ (mod 7)

3¹⁰⁰ ≡ 3⁴ = 81 (mod 7)

Find the final remainder of 81 divided by 7:

81 = 7 × 11 + 4 ⇒ Remainder = 4

Correct Option: B

💡 FundaMakers Tip:

Fermat's Little Theorem is a major time-saver for prime denominators. Dividing the exponent by (p - 1) narrows the problem down to a simple low-power evaluation like 3⁴ (mod 7) in one step.