E.g you have to find the number of trailing zeros of 28! in base 5.
How would you go about it without writing 28! in base 5? Is there a certain step by step method?
I already wrote 28! as a product of its primes.
28! = 2^25 * 3^13 * 5^6 * 7^4 * 11^2 * 13^2 * 17 * 19 * 23
The number of trailing zeros of an integer $n$ expressed in base-$b$ is equal to the greatest value of $k$ such that $b^k \mid n$, so in this case you need to find the greatest power of $5$ that divides $28!$.
By finding its entire prime factorisation, you've created far more work for yourself than you needed to!