Question

<!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> If $9^7+7^9$ is divided by $64$, then the remainder is

• A. $1$
• B. $0$
• C. $61$
• D. $11$

Answer 1

The correct option is B $0$

Given
$9^7+7^9=(1+8{)}^7-(1-8{)}^9=[1+{}^7{C}_{1}8+\dots +{}^7{C}_7{8}^7]-[1-{}^9{C}_{1}8+\dots -{}^9{C}_9{8}^9]=8[{}^7{C}_1+{}^9{C}_1]+64k,k\in Z=128+64k=64(k+2)$

Therefore, the given number is divisible by $64$, so remainder is $0$.