Slicing Using Factorizing

Given by the question,

$xy=100 = 10^{2} = (2 \times 5)^{2} = 2^{2} \times 5^{2}$

Since $x$ isn't divisible by $2$ , the only possible solutions for $x$ are either $5^{0}$ , $5^{1}$ or $5^{2}$ .

Firstly, case where $x = 5^{0} = 1$ attains us with $y =\frac{100}{1} = 100$ .

Secondly, case where $x = 5^{1}$ attains us with $y =\frac{100}{5} = 20$ .

The last case, where $x=5^{2}$ returns us with $y = \frac{100}{25} = 4$ .

Finally, our solutions are $(x,y)$ : $(1,100)$ , $(5,20)$ and $(25,4)$ .

Hence our answer is: $1+100+5+20+25+4 = \boxed{155}$ .