2010 AMC 12A Problems/Problem 23

Problem

The number obtained from the last two nonzero digits of $90!$ is equal to $n$ . What is $n$ ?

$\textbf{(A)}\ 12 \qquad \textbf{(B)}\ 32 \qquad \textbf{(C)}\ 48 \qquad \textbf{(D)}\ 52 \qquad \textbf{(E)}\ 68$

Solution

We will use the fact that for any integer $n$ , $\begin{align*}(5n+1)(5n+2)(5n+3)(5n+4)&=[(5n+4)(5n+1)][(5n+2)(5n+3)]\\ &=(25n^2+25n+4)(25n^2+25n+6)\equiv 4\cdot 6\\ &=24\pmod{25}\equiv -1\pmod{25}.\end{align*}$

First, we find that the number of factors of $10$ in $90!$ is equal to $\left\lfloor \frac{90}5\right\rfloor+\left\lfloor\frac{90}{25}\right\rfloor=18+3=21$ . Let $N=\frac{90!}{10^{21}}$ . The $n$ we want is therefore the last two digits of $N$ , or $n\pmod{100}$ . Since there is clearly an excess of factors of 2, we know that $N\equiv 0\pmod 4$ , so it remains to find $n\pmod{25}$ .

If we take out all the factors of $5$ in $N$ , we can write $N$ as $\frac M{2^{21}}$ where $M=1\cdot 2\cdot 3\cdot 4\cdot 1\cdot 6\cdot 7\cdot 8\cdot 9\cdot 2\cdots 89\cdot 18,$ where every factor of 5 is replaced by the number with all its factors of 5 removed. Specifically, every number in the form $5n$ is replaced by $n$ , and every number in the form $25n$ is replaced by $n$ .

The number $M$ can be grouped as follows:

$\begin{align*}M= &(1\cdot 2\cdot 3\cdot 4)(6\cdot 7\cdot 8\cdot 9)\cdots(86\cdot 87\cdot 88\cdot 89)\\ &\cdot (1\cdot 2\cdot 3\cdot 4)(6\cdot 7\cdot 8\cdot 9)\cdots (16\cdot 17\cdot 18) \\ &\cdot (1\cdot 2\cdot 3).\end{align*}$

Using the identity at the beginning of the solution, we can reduce $M$ to

$\begin{align*}M&\equiv(-1)^{18} \cdot (-1)^3(16\cdot 17\cdot 18) \cdot (1\cdot 2\cdot 3) \\ &= 1\cdot -21\cdot 6\\ &= -1\pmod{25}=24\pmod{25}.\end{align*}$

Using the fact that $2^{10}=1024\equiv -1\pmod{25}$ (or simply the fact that $2^{21}=2097152$ if you have your powers of 2 memorized), we can deduce that $2^{21}\equiv 2\pmod{25}$ . Therefore $N=\frac M{2^{21}}\equiv \frac {24}2\pmod{25}=12\pmod{25}$ .

Finally, combining with the fact that $N\equiv 0\pmod 4$ yields $n=\boxed{12\ \textbf{(A)}}$ .

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2010 AMC 12A Problems/Problem 23

Problem

Solution