If P is the greatest divisor of $49^{\mathrm{n}}+16 n-1$ for all $n \in \mathrm{~N}$, then the number of…
If P is the greatest divisor of $49^{\mathrm{n}}+16 n-1$ for all $n \in \mathrm{~N}$, then the number of factors of P is
- $12$
- $15$
- $7$
- $13$
Solution
$\begin{aligned}
& 49^n+16 n-1=(1+48)^n+16 n-1 \\
= & 1+{ }^n C_1(48)+{ }^n C_2(48)^2+\ldots .+{ }^n C_n(48)^n+16 n-1 \\
= & (48 n+16 n)+{ }^n C_2(48)^2+\ldots .+{ }^n C_n(48)^n \\
= & 64 n+8^n\left({ }^n C_2 6^2+{ }^n C_3 6^3 8+\ldots+{ }^n C_n 6^n 8^{n-2}\right)
\end{aligned}$
Hence, $49^{\mathrm{n}}+16 n-1$ is divisible by 64 and factors of 64 are $1,2,4,8,16,32,64$. So, number of factors $=7$
Asked in: AP EAMCET 2024 (22 May Shift 2)
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