Question

When $2^{155}$ is divided by $9,$ the remainder is :

• A. $8$
• B. $7$
• C. $5$
• D. $6$
• E. $1$

Answer 1

Answer: (C)

$5$

$2^{155}=(2^6{)}^{25}\times 2^5$
$=(64{)}^{25}\times 32$
$=(1+63{)}^{25}\times 32$
$={[}^{25}{C}_0\left(63{)}^0{+}^{25}{C}_1\right(63{)}^1{+}^{25}{C}_2(63{)}^2+...{+}^{25}{C}_{25}(63{)}^{25}]32$
$=[1+(63\left){+}^{25}{C}_2\right(63{)}^2+...+\left(63{)}^{25}\right]32$
$=32+63\left[1{+}^{25}{C}_2\right(63)+...+(63{)}^{24}]32$
The remainder when $2^{155}$ is divided by $9$
The remainder when $32$ divided by $9=5$.