Question

The hundred's digit of $3^{100}$ is

• A. $0$
• B. $1$
• C. $2$
• D. $7$

Answer 1

The correct option is A $0$

$3^{100}$
$=(9{)}^{50}$
$=(10-1{)}^{50}$
$={10}^{50}-{}^{10}{C}_1{10}^{49}+{}^{10}{C}_2{10}^{48}-...-{}^{50}{C}_{49}10+{}^{50}{C}_{50}{10}^0$
$=1000\left(m\right)+{}^{50}{C}_{48}{10}^2-{}^{50}{C}_{49}10+{}^{50}{C}_{50}{10}^0$
$=1000m+\frac{50.49}{2}{10}^2-50\left(10\right)+1$
$=1000\left(m\right)+\left(122500\right)-500+1$
$=1000\left(m\right)+122000+1$
$=1000\left(m\right)+122001$
Hence the digit at hundredth place is 0.