Number Theory
Remainder by binomial theorem
MMTS_Full_Test_15
Grade 12
Question:
Remainder when $(2022)^{2023}$ is divided by 11 is
Step-by-Step Solution
Key Concept: $2022=11\cdot183+9\equiv9\equiv-2\pmod{11}$. $(-2)^{2023}=-2^{2023}$. $2^{10}\equiv1\pmod{11}$ (Fermat). $2023=10\cdot202+3$. $2^{2023}\equiv2^3=8\pmod{11}$. $-8\equiv3\pmod{11}$.
Remainder $=3$.
Correct Answer: 3