Question

$7^{9}+9^{7}$ is divisible by

• A. 16
• B. 24
• C. 64
• D. 72

Answer 1

Answer: (C)

We have,
$7^{9}+9^{7} =(1+8)^{7}-(1-8)^{9} $
$=\left(1+{ }^{7} C_{1} \cdot 8^{1}+{ }^{7} C_{2} \cdot 8^{2}+\ldots+{ }^{7} C_{7} \cdot 8^{7}\right) $
$-\left(1-{ }^{9} C_{1} \cdot 8^{1}+{ }^{9} C_{2} \cdot 8^{2}-\ldots{ }^{9} C_{9} \cdot 8^{9}\right) $
$=16 \times 8+64\left[\left({ }^{7} C_{2}+\ldots+{ }^{7} C_{7} \cdot 8^{5}\right)\right.$
$\left.-\left({ }^{9} C_{2}-\ldots-{ }^{9} C_{9} \cdot 8^{7}\right)\right] $
$=64 $ (an integer)
Hence, $7^{9}+9^{7}$ is divisible by $6 4$.