Let me first pose the questions free of context:
Given prime $p$ and positive integers $b$ and $N$, define $F(p,b,N) = \Big\lfloor (1/b) \sum_{i=1}^\infty \lfloor N/p^i \rfloor \Big\rfloor.$
That is, $F(p,b,N)$ is the exponent of $p$ in the prime factorization of $N!$, divided by $b$, and the quotient truncated to the greatest integer.
Let $\hat F$ approximate $F$ by dropping all use of the floor function: $\hat F(p,b,N) = (1/b) \sum_{i=1}^\infty N/p^i = N/b(p-1).$
The question is this: Given various $p$ and $b$, how valid is it to use $\hat F(p,b,1)$ (that is, ignoring $N$) as a proxy for comparing $F$ itself over all $N$? Specifically:
Conjecture 1. Suppose $p_1$, $p_2$ are primes and $b_1$, $b_2$ are positive integers such that $\hat F(p_1, b_1, 1) < \hat F(p_2, b_2, 1)$. Then $F(p_1,b_1,N) \leq F(p_2,b_2,N)$ for all positive integers $N$.
Conjecture 2. Suppose $p_1, p_2, \ldots, p_k$ are distinct primes and $b_1, b_2, \ldots, b_k$ are positive integers such that $\hat F(p_i, b_i, 1) = \hat F(p_j, b_j, 1)$ for all $1 \leq i, j \leq k$. Then there are infinitely many positive integers $N$ such that, for all $1 < i \leq k$, we have $F(p_1,b_1,N) < F(p_i,b_i,N)$.
You may ask: Out of what blue did these conjectures arise? The context is a not-very-deep discussion of how to calculate the number of trailing zeros of $N!$ written in various bases. You can see it, including the use of these conjectures, here: http://denenberg.com/fzeros.pdf .