Binomial Theorem
Divisibility Using Binomial Theorem
nta_pyq_2023_apr
Grade None
Question:
Among the statements: (S1): $2023^{2022}-1999^{2022}$ is divisible by $8$. (S2): $13(13)^n-11n-13$ is divisible by $144$ for infinitely many $n\in\mathbb{N}$.
Only (S2) is correct
Only (S1) is correct
Both (S1) and (S2) are correct
Both (S1) and (S2) are incorrect
Step-by-Step Solution
Key Concept: (S1): $(a^n-b^n)$ is divisible by $(a-b)$. Here $2023-1999=24$, divisible by 8. (S2): Expand $13(1+12)^n$ via binomial; remainder after simplification is $145n+144\lambda$, not divisible by 144 in general.
(S1): $24|2023-1999$ and $8|24$, so true. (S2): $13(13)^n-11n-13=145n+144k$; for divisibility by 144 need $144|145n$, i.e., $144|n$. Not for infinitely many general $n$. Only (S1) is correct.
Correct Answer: 2