This question is from an old exam in a problem-solving course.
Show that there exists $20 \leq m, n \leq 5000$ so that the starting six digits of $2^m$ and $3^n$ are within 1 of each other.
I'm not entirely sure where to begin with this question. I'm definitely thinking there should be a number theory argument here using modular arithmetic but not sure how to construct one. The starting 6 digits imply that the powers would have to be very large and to examine those digits I guess, I'd be looking at the values mod 10 but I'm not certain how to begin such an approach.
Any assistance with this problem would be appreciated!