Question

The unit digit of ${17}^{1983}+{11}^{1983}-7^{1983}$ is

• A. $1$
• B. $2$
• C. $3$
• D. $0$

Answer 1

The correct option is A $1$

The above expression can be rewritten as
$(20-3{)}^{1983}+(1+10{)}^{1983}-(10-3{)}^{1983}$
Let $n=1983$
Therefore
$(20-3{)}^n+(1+10{)}^n-(10-3{)}^n$
$=({20}^n-3{}^n{C}_1{20}^{n-1}...-3^n)+(1+{}^n{C}_{1}10+n{C}_2{10}^2..{.10}^n)-({10}^n-3{}^n{C}_1{10}^{n-1}...-3^n)$
Now in the above expansion.
The units digits will be given by
$-3^n+1-(-3^n)$
$=1$