2010 AMC 10A Problems/Problem 24

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 1(Bigbrain)

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 $N\pmod{25}$ .

We can write $N$ as $\frac M{2^{21}}$ where $M=1\cdot 2\cdot 3\cdot 4\cdot 1\cdot 6\cdot 7\cdot 8\cdot 9\cdot 2\cdots 89\cdot 18 = \frac{90!}{5^{21}},$ where every number in the form $5n$ is replaced by $n$ .

The number $M$ can be grouped as follows:

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

Hence, 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 $2^{10}=1024\equiv -1\pmod{25}$ ,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 $n=\boxed{\textbf{(A)}\ 12}$ .

Solution 2(bash)

First, we list out all the numbers
$90!=2^{65}3^{44}5^{21}7^{13}11^813^617^519^423^329^331^237^241^243^2\cdot47\cdot53\cdot\cdot\cdot89$ Since we must get rid of ending $0$ s, we get rid of $5^21$ and the corresponding $2^21$ Next, we note that $2^22\equiv2^2$ , $3^20\equiv3$ , and $7^4\equiv7$ , so it can be simplified to $2^43^47\cdot11^813^617^5\cdot\cdot\cdot89$

2010 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 23	Followed by Problem 25
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2010 AMC 10A Problems/Problem 24

Contents

Problem

Solution 1(Bigbrain)

Solution 2(bash)

See also