Question

The remainder, when \(7^{98}\) is divided by 23, is equal to:

• A. 14
• B. 9
• C. 17
• D. 6

Hint:

For modular arithmetic, Fermat's theorem is a powerful tool for simplifying large exponents.

Answer 1

The Correct Option is A

Step 1: Using Fermat's theorem.
Since 23 is prime, by Fermat’s theorem: \[ 7^{22} \equiv 1 \pmod{23} \] Now, break down the exponent: \[ 7^{98} = 7^{(22 \times 4) + 10} = 7^{22 \times 4} \times 7^{10} \equiv 1^4 \times 7^{10} = 7^{10} \pmod{23} \] Calculating \( 7^{10} \): \[ 7^{10} = 28 \times 25 \equiv 14 \pmod{23} \]