<s>[INST] Find the last two non-zero digits of $70!$ The question itself is quite straightforward, however, I am unable to get an exact answer to the problem. I have narrowed it down to four possibilities one from $\{18, 43, 68, 93\}$.
The approach
We begin by finding the number of zeros that are contained in $70!$. This is done as shown below:
$$
\sum_{n=1}^\infty \left\lfloor \frac{70}{5^n} \right\rfloor = 16
$$
And clearly
$$
\sum_{n=1}^\infty \left\lfloor \frac{70}{2^n} \right\rfloor > \sum_{n=1}^\infty \left\lfloor \frac{70}{5^n} \right\rfloor \qquad \qquad \text{(i)}
$$
So therefore we define some integer $N$ such that:
$$
N = \frac{70!}{5^{16}2^{16}}
$$
From here we will find the remainder of $N$ modulo $4$ and $25$ respectively so we can use the Chinese Remainder Theorem to finish the problem.
Verifying mod 4
From (i) it is clear that $N$ will enough spare twos to be divisible by $4$. This leads us to the conclusion that:
$$
N \equiv 0 \pmod 4
$$
Verifying mod 25
We can split this part of the problem into two parts. Define another integer $M$ such that
$$
M = \frac{70!}{5^{21}}
$$
This allows us to write:
$$
\frac{1}{2^{16}} \cdot M = N
$$
Finding 2^16 mod 25
Trivially, we can find that
$$
2^{16} = 65536 \equiv 11 \pmod {25}
$$
Finding M mod 25
This one is a little trickier. To do this, we will use the fact that:
$$
(5n+1)(5n+2)(5n+3)(5n+4) \equiv -1 \pmod {25}
$$
Notice that we can group $M$ as follows:
$$
\begin{align*}
(1\cdot2\cdot3\cdot4)(6\cdot7\cdot8\cdot9)\cdots(66\cdot67\cdot68\cdot69) \ &\times \\
(1\cdot2\cdot3\cdot4)(6\cdot7\cdot8\cdot9)(11\cdot12\cdot13\cdot14) \ &\times \\
(1\cdot2)
\end{align*}
$$
Now, we can easily see:
$$
M \equiv (-1)^{12} \cdot (-1)^{3} \cdot {2} \equiv 23 \pmod {25}
$$
Bringing it together
Now we have the following equation
$$
N \equiv \frac{M}{2^{16}} \equiv \frac{23}{11} \pmod {25}
$$
This is where I got stuck.
From here I tried to write
$$\frac{23 + 25k}{11} \pmod{25}$$
and this yields four possible solutions $\{18, 43, 68, 93\}$. Is there a way that I can find the correct solution and/or a better way to solve the problem?
Thanks!
 [/INST] 
From here we will find the remainder of $N$ modulo $4$ and $25$
respectively so we can use the Chinese Remainder Theorem to finish the
problem.
$$ N \equiv 0 \pmod 4 $$
$$ N \equiv \frac{M}{2^{16}} \equiv \frac{23}{11} \pmod {25} $$
$$\color{red}{\text{THIS IS WHERE I GOT STUCK!}}$$

To simplify $\frac{23}{11} \pmod {25}$, you can compute the inverse of 11 mod 25, by solving the Diophantine equation $11 x - 25 q = 1$. This will give you $11^{-1} = 16 \pmod {25}$. Thus $N \equiv 23 \times 16 \equiv 18 \pmod{25} $.
Now you can apply your last planned step, which was to use the Chinese remainder theorem to go from $N \equiv 0 \pmod{4}, N \equiv 18 \pmod{25}$ to $N \equiv ?? \pmod{100}$.
But actually, since $100 = 25 \times 4$, we don't even need the heavy artillery here. Just list the 4 possibilities $\{18, 43, 68, 93\}$ and cross out the 3 which are not $\equiv 0 \pmod{4}$.
We end up with $N \equiv 68 \pmod{100}$.
My computer confirms that result:
$$70! = 11978571669969891796072783721689098736458938142546425857555362864628009582789845319\color{red}{68}0000000000000000$$
</s>