Number of Divisors

One could calculate that

$7^{10}+7^{11}+7^{12}+7^{13}=7^{10}(1+7+7^{2}+7^{3})=7^{10}*400= 2^{4}*5^{2}*7^{10}$

To count the positive divisors of an integer, one only need to multiply all of the powers of each prime factors, after adding each of them by $1$ . Thus, this leads us to $5*3*11=165$ .

=7^10 2^4 5^2 Numbers of divisors=11×5×3= 165

for those who likes to code ,this is a simple but smooth solution in Mathematica

1

Length[Divisors[112990099600]]

what is doing, first $112990099600 = 7^{10} + 7^{11} + 7^{12} + 7^{13}$

the I'm calling a the function Divisors[112990099600] that return a list with all the divisor of that number, but in this problem I do not need to numerate them, so then I apply the Length[...] function to that list, that returns me how many elements does it have, and that gives me $165$