Question

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

• A. 16
• B. 24
• C. 64
• D. 72

Answer 1

Answer: (C)

We have,
$7^9+9^7=(1+8{)}^7-(1-8{)}^9$
$=\left(1+{}^7{C}_1\cdot 8^1+{}^7{C}_2\cdot 8^2+\dots +{}^7{C}_7\cdot 8^7\right)$
$-\left(1-{}^9{C}_1\cdot 8^1+{}^9{C}_2\cdot 8^2-\dots {}^9{C}_9\cdot 8^9\right)$
$=16\times 8+64[\left({}^7{C}_2+\dots +{}^7{C}_7\cdot 8^5\right)$
$-\left({}^9{C}_2-\dots -{}^9{C}_9\cdot 8^7\right)]$
$=64$ (an integer)
Hence, $7^9+9^7$ is divisible by $64$.