Number Theory
Remainder by modular arithmetic
MJMT_Full_Test_04
Grade 12
Question:
Let $X=(23\sqrt3+39)^{2025}$, $Y=X-[X]$ (GIF). Remainder when $XY$ is divided by 31 is
Step-by-Step Solution
Key Concept: $(23\sqrt3+39)^{2025}+(23\sqrt3-39)^{2025}$ is an integer (conjugate pair). Let $XY=X(X-[X])$. Use binomial: $(23\sqrt3+39)^{2025}+(39-23\sqrt3)^{2025}\equiv\ldots\pmod{31}$.
Remainder $=1$.
Correct Answer: 1