Binomial Theorem
Remainders of Power Expressions
nta_pyq_2023_apr
Grade 11
Question:
Let the number $(22)^{2022}+(2022)^{22}$ leave the remainder $\alpha$ when divided by $3$ and $\beta$ when divided by $7$. Then $(\alpha^2+\beta^2)$ is equal to
Step-by-Step Solution
Key Concept: For mod 3: $22\equiv 1\pmod{3}$, so $22^{2022}\equiv 1$; $2022\equiv 0\pmod{3}$, so $2022^{22}\equiv 0$. Total $\equiv 1\Rightarrow\alpha=1$. For mod 7: Use $(21+1)^{2022}+(2023-1)^{22}$.
$\alpha=1,\ \beta=2$. $\alpha^2+\beta^2=5$.
Correct Answer: 3