Suppose the original 5 digit number was $10x+y$ where $x$ is a four digit number and $y$ is the units digit of the 5 digit number. We know that $10x+y+x=11x+y=52713$. Because $x$ is a 4 digit number and $y$ is a 1 digit number, we can try all values 0-9 for $y$ and get different 4 digit values for $x$ then just find all possible sums of the digits for $10x+y$. Don't know if there is a faster solution.

well there is only one value of y such that x is an integer and that is when y=1 when $y = 1$, and we get $x = 4792$ there fore the sum of the five digits is $4+7+9+2+1 = 23$