Question

The remainder when 7¹⁰³ is divided by 17, is

Answer 1

Correct Answer: 12

We are given \( 7^{103} \) and asked to find the remainder when divided by 17. First, express \( 7^{103} \) as: \[ 7^{103} = 7 \times 7^{102}. \] We can apply modulo 17 and reduce the powers step by step: \[ 7^{102} = (7^2)^{51} \quad \Rightarrow \quad 7^2 = 49 \equiv 15 \pmod{17}. \] Now, calculate: \[ 7^{102} = 15^{51} \equiv (-2)^{51} \equiv -2^{51} \pmod{17}. \] By reducing powers of 2 modulo 17, we find that \( 2^{51} \equiv 2 \pmod{17} \). Hence: \[ 7^{102} \equiv -2 \pmod{17} \quad \Rightarrow \quad 7^{103} = 7 \times (-2) \equiv -14 \equiv 12 \pmod{17}. \] Thus, the remainder when \( 7^{103} \) is divided by 17 is 12.