>>10961605$10^{40} = 2^{40}*5^{40}$ and $20^{30} = 4^{30}*5^{30} = 2^{60}*5^{30}$
To find common factors:-
Note,
1) All common factors are of the form 2^n*5^m, where n,m are integers $\geq 0$
2) if $n>40$ then not divisor for $10^{40}$, if $m>30$ then not divisor for $20^{30}$.
So we got $0 \leq n \leq 40$ and $0 \leq m \leq 30$
that is total of 41*31 = 41 * 31
1271 possibilities,
>>10961841, as anon pointed out.
