COMEDK · Maths · 7. Binomial Theorem
If \(49^n + 16n + k\) is divisible by 64 for \(n \in \mathbb{N}\), then the greatest negative integral value of \(k\) is
- A \(-\)1
- B \(-\)4
- C \(-\)3
- D \(-\)2
Answer & Solution
Correct Answer
(A) \(-\)1
Step-by-step Solution
Detailed explanation
\(49^n = (1+48)^n = 1 + 48n + \binom{n}{2}(48)^2 + \ldots\) Since \(48^2 = 2304 = 64 \times 36\), all terms from \(\binom{n}{2}(48)^2\) onwards are divisible by 64. \(49^n \equiv 1 + 48n \pmod{64}\)…
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