Binomial Theorem
Divisibility of Sum of Powers
nta_pyq_2023_apr
Grade 11
Question:
$25^{190}-19^{190}-8^{190}+2^{190}$ is divisible by
neither 14 nor 34
14 but not by 34
34 but not by 14
both 14 and 34
Step-by-Step Solution
Key Concept: Group as $(25^{190}-19^{190})-(8^{190}-2^{190})$: each bracket is divisible by $25-19=6$ and $8-2=6$ — not helpful. Regroup: $(25^{190}-8^{190})-(19^{190}-2^{190})$ divisible by $17$. Also divisible by $2$. Check divisibility by $7$.
Divisible by 34 but not 14.
Correct Answer: 3