Question

The remainder when $3^{100}$ is divided by $100$ is

• A. $01$
• B. $21$
• C. $41$
• D. $61$

Answer 1

The correct option is A $01$

Let $p=3^{100}=9^{50}$
$=(10-1{)}^{50}={}^{50}{C}_0\cdot {10}^0\cdot (-1{)}^{50}+{}^{50}{C}_1\cdot {10}^1\cdot (-1{)}^{49}+{}^{50}{C}_2\cdot {10}^2\cdot (-1{)}^{48}+\cdots =1-500+{}^{50}{C}_2\times 100+100m\left(\text{where m is a +ve integer}\right)=1+100({}^{50}{C}_2+m-5)=1+100k$

So, the remainder is $01.$