Q117 / 180 · COMEDK 2014 · Mathematics

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

  • A
    128
  • B
    24
  • Correct answer
    C
    64
  • D
    72

Solution

Now, $7^{9}=(8-1)^{9}=-1(1-8)^{9}$ $=-1+{ }^{9} C_{1} 8-{ }^{9} C_{2} 8^{2}+\ldots+{ }^{9} C_{9} 8^{9}$ and $\quad 9^{7}=(1+8)^{7}=1+{ }^{7} C_{1} 8+{ }^{7} C_{2} 8^{2}$ $+{ }^{7} C_{3} 8^{3}+\ldots .+{ }^{7} C_{7} 8^{7}$ $\therefore 7^{9}+9^{7}=8\left({ }^{9} C_{1}+{ }^{7} C_{1}\right)+8^{2}\left({ }^{7} C_{2}-{ }^{9} C_{2}\right)+\ldots$ $=8(9+7)+8^{2}(21-36)+\ldots$ $=64 \times 2+64(-15)+\ldots$ Hence, it is divisible by 64 .

Hence, the correct answer is option: 'C'.

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