Question

The remainder, when \( 15^{23} + 23^{23} \) is divided by 19, is

• A. 4
• B. 15
• C. 0
• D. 18

Hint:

In problems involving large exponents and modular arithmetic, reduce the base modulo the divisor to simplify the calculation.

Answer 1

The Correct Option is C

We need to find the remainder when \( 15^{23} + 23^{23} \) is divided by 19. First, reduce \( 15 \) and \( 23 \) modulo 19: \[ 15 \equiv -4 \pmod{19}, \quad 23 \equiv 4 \pmod{19} \] Thus, we need to find the remainder when \( (-4)^{23} + 4^{23} \) is divided by 19. Since \( (-4)^{23} = -4^{23} \), we have: \[ (-4)^{23} + 4^{23} = 0 \] Therefore, the remainder when \( 15^{23} + 23^{23} \) is divided by 19 is \( 0 \). \[ \boxed{0} \]