$49^{\mathrm{n}}+16 \mathrm{n}-1$ is divisible by

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Professor PC Thomas Classes · Answer

$ \begin{aligned} & 49^n+16 n-1=(1+48)^n+16 n-1 \ & =1+48 n+\ldots .+(48)^n+16 n-1 \ & =64 n+{ }^n C_2(48)^2+{ }^n C_3(48)^3+ \ & \quad \ldots .+(48)^n . \ & =64\left\{n+{ }^n C_2 6^2+{ }^n C_3 6^3 \cdot 48+\ldots+6^n 8^{n-2}ight\} \ & 	herefore 49^n+16 n-1 \end{aligned} $ \  is divisible by 64 .