2020 AMC 12A Problems/Problem 21

Problem 21

How many positive integers $n$ are there such that $n$ is a multiple of $5$ , and the least common multiple of $5!$ and $n$ equals $5$ times the greatest common divisor of $10!$ and $n?$

$\textbf{(A) } 12 \qquad \textbf{(B) } 24 \qquad \textbf{(C) } 36 \qquad \textbf{(D) } 48 \qquad \textbf{(E) } 72$

Solution

We set up the following equation as the problem states:

$\text{lcm}{(5!, n)} = 5\text{gcd}{(10!, n)}.$

Breaking each number into its prime factorization, we see that the equation becomes

$\text{lcm}{(2^3\cdot 3 \cdot 5, n)} = 5\text{gcd}{(2^8\cdot 3^4 \cdot 5^2 \cdot 7, n)}.$

We can now determine the prime factorization of $n$ . We know that its prime factors belong to the set $\{2, 3, 5, 7\}$ , as no factor of $10!$ has $11$ in its prime factorization, nor anything greater. Next, we must find exactly how many different possibilities exist for each.

There can be anywhere between $3$ and $8$ $2$ 's and $1$ to $4$ $3$ 's. However, since $n$ is a multiple of $5$ , and we multiply the $\text{gcd}$ by $5$ , there can only be $3$ $5$ 's in $n$ 's prime factorization. Finally, there can either $0$ or $1$ $7$ 's.

Thus, we can multiply the total possibilities of $n$ 's factorization to determine the number of integers $n$ which satisfy the equation, giving us $6 \times 4 \times 1 \times 2 = \boxed{\textbf{(D) } 48}$ . ~ciceronii

Solution 2

Like the Solution 1, we starts from the equation:

$\text{lcm}{(5!, n)} = 5\text{gcd}{(10!, n)}.$ Assume $\text{lcm}{(5!, n)}=k\cdot5!$ , with some integer $k$ . It follows that $k\cdot 4!=\text{gcd}{(10!, n)}$ . It means that $n$ has a divisor $4!$ . Since $n$ is a multiple of $5$ , $n$ has a divisor $5!$ . Thus, $\text{lcm}{(5!, n)}=n=k\cdot5!$ . The equation can be changed as $k\cdot5!=5\text{gcd}{(10!, k\cdot5!)}$ $k=5\text{gcd}{(6\cdot7\cdot8\cdot9\cdot10, k)}$ We can see that $k$ is also a multiple of $5$ , with a form of $5\cdot m$ . Substituting it in the above equation, we have $m=5\text{gcd}{(6\cdot7\cdot8\cdot9\cdot2, m)}$ Similarly, $m$ is a multiple of $5$ , with a form of $5\cdot s$ . We have $s=\text{gcd}{(6\cdot7\cdot8\cdot9\cdot2, 5\cdot s)}=\text{gcd}{(2^5\cdot3^3\cdot7, s)}$ The equation holds, if $s$ is a divisor of $2^5\cdot3^3\cdot7$ , which has $(5+1)(3+1)(1+1)=\boxed{(\text{D})48}$ divisors.

by Linty Huang

2020 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 20	Followed by Problem 22
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2020 AMC 12A Problems/Problem 21

Problem 21

Solution

Solution 2