Binomial Theorem
Remainder Using Binomial Theorem
nta_pyq_2023_apr
Grade 11
Question:
The remainder, when $7^{103}$ is divided by $17$, is
Step-by-Step Solution
Key Concept: Write $7^{103}=(7^2)^{51}\cdot 7=(51-2)^{51}\cdot 7$. Expand $(51-2)^{51}\pmod{17}$: only the last term survives since $51=3\times17\equiv0\pmod{17}$.
$7^{103}\equiv 7\cdot(-2)^{51}\equiv-56\equiv 12\pmod{17}$.
Correct Answer: 12