Question

The expression $7^9+9^7$ is divisible by $64$. State true or false.

Answer 1

Given: $7^9+9^7$

$(a+b{)}^n{=}^n{C}_0{a}^n{+}^n{C}_1{a}^{n-1}b+\dots {+}^n{C}_n{b}^n$

$\therefore (1+x{)}^n{=}^n{C}_0{+}^n{C}_{1}x+\dots {+}^n{C}_n{x}^n$

Now,

$7^9+9^7=(1+8{)}^7-(1-8{)}^9$

Use the expansion of $(1+x{)}^n$

$\Rightarrow 7^9+9^7=[1{+}^7{C}_{1}8{+}^7{C}_2{8}^2+\dots {+}^7{C}_7{8}^7]-[1{-}^9{C}_{1}8{+}^9{C}_2{8}^2-\dots {-}^9{C}_9{8}^9]$

$\Rightarrow 7^9+9^7=(1-1)+(7\times 8)+(9\times 8)+$

$8^2{(}^7{C}_2{-}^9{C}_2{+}^7{C}_{3}8{+}^9{C}_{3}8+\dots {+}^9{C}_9{8}^7)$

$\Rightarrow 7^9+9^7=8\times (7+9)+64\lambda ,\lambda \in N$

$\Rightarrow 7^9+9^7=8\times 16+64\lambda =64(\lambda +2)$

Therefore, it is a multiple of $64$.

Hence, the given statement is True.