Question

If $6^n-5n,n\in N$ is divided by $25$, then the remainder is

• A. $1$
• B. $3$
• C. $5$
• D. $0$

Answer 1

The correct option is A $1$

$6^n-5n=(1+5{)}^n-5n=(1+{}^n{C}_1\cdot 5+{}^n{C}_2\cdot 5^2+{}^n{C}_3\cdot 5^3+\dots )-5n=(1+5n+{}^n{C}_2\cdot 5^2+{}^n{C}_3\cdot 5^3+\dots )-5n=1+5n+25({}^n{C}_2+{}^n{C}_3\cdot 5+\dots )-5n=1+5n+25k-5n=25k+1$

Hence, the remainder is $1$.


Alternate solution:
Putting $n=2$, we get
$6^2-5\times 2=26$
So, when divided by $25$, we get remainder as $1$.
Putting $n=3$, we get
$6^3-5\times 3=216-15=201$
So, when divided by $25$, we get remainder as $1$.

Hence, the required remainder is $1$.