Question

What is the sum of digits of the number ${2013}^{2013}$ (^ is raised to the power).

Answer 1

${2013}^{2013}$
Sum of digits = Digital Sum
Method followed is the remainder when number is divided by 9.
When a number is divided by 9, the remainder gives the digital sum.
Example $\frac{47}{9}$ = Remainder = 2
and Digital sum (47) = 4+7 = 11 = 2.
Similarly or Applying the above method $=\frac{(2013{)}^{2013}}{9}=\frac{6\times 6\times 6\times ...\times 6\to Remainder(6-2013times)}{9}$
$=\frac{(6\times 6\times 6\times ...\times 6(2012times\left)\right)\times 6}{9}$
For 2012 times 6, it is divided by 9 - remainder is 0 since $6^2=36$.
Number left = 6
Hence Remainder = 6.
$\therefore$ Digital Sum = 6.