Question

The remainder of ${51}^{51}$, when divided by $25$ is :

• A. $0$
• B. $1$
• C. $2$
• D. None of the above.

Answer 1

The correct option is B $1$

${51}^{51}$ can be written as $(1+50{)}^{51}$

On expanding, $1{+}^{51}{C}_1\left(50\right){+}^{51}{C}_2{\left(50\right)}^2{+}^{51}{C}_3{\left(50\right)}^3+....{+}^{51}{C}_{51}{\left(50\right)}^{51}=1+50({}^{51}{C}_1{+}^{51}{C}_2\left(50\right){+}^{51}{C}_3{\left(50\right)}^2+....{+}^{51}{C}_{51}{\left(50\right)}^{50})=1+25k$

Hence, Remainder$=1$ when divided by 25.