Question

When $2^{1505}$ is divided by $9$, the remainder is

• A. 8
• B. 7
• C. 5
• D. 6
• E. 1

Answer 1

Answer: (C)

$2^{1505}={\left(2^6\right)}^{25}\times 2^5$
$=(64{)}^{25}\times 32$
$=(1+63{)}^{25}\times 32$
$=[{}^{25}{C}_0(63{)}^0+{}^{25}{C}_1(63{)}^1+{}^{25}{C}_2(63{)}^2$
$\dots .+{}^{25}{C}_{25}(63{)}^{25}]32$
$=\left[1+(63)+{}^{25}{C}_2(63{)}^2+\dots +(63{)}^{25}\right]32$
$=32+63\left[1+{}^{25}{C}_2(63)+\dots +(63{)}^{24}\right]32$
The remainder when $2^{1505}$ is divided by
9 $=$ The remainder when 32 divided by 9
$=5$