Question

$7^9+9^7$ is divided by

• A. $128$
• B. $24$
• C. $64$
• D. $72$

Answer 1

The correct option is C

$64$

Explanation for the correct option:

Step1.Use the binomial expansion to expand the expression.

$\begin{array}{rcl}{7}^9+9^7& =& {\left(8-1\right)}^9+{\left(8+1\right)}^7\\ & =& \left[{}^{7}C_0+{}^{7}C_1·8+{}^{7}C_2·{\left(8\right)}^2...+{}^{7}C_7·{\left(8\right)}^7\right]-\left[{}^{9}C_0-{}^{9}C_1·8+{}^{9}C_2·{\left(8\right)}^2...-{}^{9}C_9·{\left(8\right)}^9\right]\\ & =& 8^{2}k+72-1+56+1\\ & =& 64k+128\end{array}$ $\left[\mathbf{\because }{\left(a+b\right)}^{\mathbf{n}}\mathbf{=}{}^{\mathbf{n}}\mathit{C}_{\mathbf{0}}{\mathit{a}}^{\mathbf{n}}{\mathit{b}}^{\mathbf{0}}\mathbf{+}{}^{\mathbf{n}}\mathit{C}_{\mathbf{1}}{\mathit{a}}^{\mathbf{n}\mathbf{-}\mathbf{1}}{\mathit{b}}^{\mathbf{1}}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}{}^{\mathbf{n}}\mathit{C}_{\mathbf{n}}{\mathit{a}}^{\mathbf{0}}{\mathit{b}}^{\mathbf{n}}\right]$

Step 2: Find the common factor from the expression:

Find the value that is common in both the terms of the expression on the right-hand side of the equation.

Take $64$ common from the expression.

$7^9+9^7=64(k+2)$

So, the given expression is divisible by $64$.

Hence, option (C) is the correct answer.