For each integer $n$, $p(n)$ outputs an ordered pair whose members are remainders of $n$ when divided by $4$ and $6$ respectively.
So
$$p(n) = (n \mod 4, n \mod 6).
$$
If ten thousand integers are chosen at random, how many can you say for certain must have the same value for $p$?
I did a similar example where instead of $4$ and $6$, it was $3$ and $4$.
The solution was just $\lceil \frac{10000}{3\times 4}\rceil = 834$.
However, my solution says it is not the same methodology for when it is $4,6$. How come? How would I do this?
(I determined the easier solution by looking at residues $\mod 3$ and residues $\mod 4$.)