I have in mind a number, that is the smallest non-zero number that:
when doubled it is a square number,
when tripled it is a cube number,
when multiplied by 5 it is a power of 5, and
when multiplied by 7 it is a power of 7.
How many zeroes does it have after the last non-zero digit?
Although this puzzle might involve very large numbers, it can be approached without a computer or even a calculator.