Let's say that $n$ and $m$ are two very large natural numbers, both expressed as product of prime factors, for example:
$n = 3×5×43×367×4931×629281$
$m = 8219×138107×647099$
Now I'd like to know which is smaller. Unfortunately, all I have is an old pocket calculator that can show at most (say) ten digits. So while there are enough digits to enter each factor individually, $n$ and $m$ are both too large to fit in the calculator. To my disappoint, they are also so close that even their logarithms are indistinguishable looking at the first 10 digits.
Question: how would one go to determine which one of two integers is smaller in a case like this? Any easier alternative than calculating the full decimal expansion of both products with pen and paper?