TS EAMCET · Maths · Basic of Mathematics
If \(2^{4 n+3}+3^{3 n+1}\) is divisible by P for all natural numbers \(n\), then P is
- A an even integer
- B an odd integer, not a prime
- C an odd prime integer
- D an integer less than 9
Answer & Solution
Correct Answer
(C) an odd prime integer
Step-by-step Solution
Detailed explanation
\( f(n) = 2^{4n+3}+3^{3n+1} \) \( f(1) = 2^{4(1)+3}+3^{3(1)+1} = 2^7+3^4 = 128+81 = 209 \) \( 209 = 11 \cdot 19 \) \( 2^{4n+3}+3^{3n+1} = 8 \cdot (2^4)^n + 3 \cdot (3^3)^n = 8 \cdot 16^n + 3 \cdot 27^n \) \( \equiv 8 \cdot 5^n + 3 \cdot 5^n \pmod{11} \)…
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