Q. If $9^{7}+7^{9}$ is divisible by $2^{n},$ then find the greatest value of $n,$ where $n\in N:-$

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A

$5$

B

$6$

C

$7$

D

None of these

Solution:

We have,
$9^{7}+7^{9}=\left(1 + 8\right)^{7}-\left(1 - 8\right)^{9}$
$=\left[\_{}^{7}C_{0}^{} + \_{}^{7}C_{1}^{} 8^{1} + \_{}^{7}C_{2}^{} 8^{2} + \ldots + \_{}^{7}C_{7}^{} 8^{7}\right]-\left[\_{}^{9}C_{0}^{} - \_{}^{9}C_{1 }^{} 8^{1} + \_{}^{9}C_{2 }^{} 8^{2} - \ldots - \_{}^{9}C_{9}^{} 8^{9}\right]$
$=128+64\left[\right.$ Integer $\left]\right.$
$=64k$ $\left(\right.$ where $k$ is some integer $\left.$
So, $n=6$ $\left[\because 2^{6} = 64\right]$ .

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