Binomial Theorem
Remainder via Binomial Theorem
nta_pyq_2025_apr
Grade 11
Question:
The remainder, when $7^{103}$ is divided by 23, is equal to:
Step-by-Step Solution
Key Concept: Write $7^3 = 343 = 15 \times 23 - 2$, express $7^{103}$ in terms of $(23k - 2)^{34}$, then expand using Binomial Theorem keeping only remainder terms.
$7^{103} = 7(343)^{34} = 7(345-2)^{34} = 23K_1 + 7 \cdot 2^{34}$. Now $7 \cdot 2^{34} = 28 \cdot (256)^4 = 28(253+3)^4 = 23K_2 + 28 \cdot 81$. And $28 \times 81 = (23+5)(69+12) \equiv 5 \times 12 = 60 \equiv 60 - 2 \times 23 = 14 \pmod{23}$. Remainder $= 14$.
Correct Answer: 14