Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2023 AMC 10B Problems/Problem 15"

(→‎Solution)
(→‎Solution)
Line 18: Line 18:
 
\end{align*}
 
\end{align*}
 
</cmath>
 
</cmath>
 +
 +
 +
== Solution 2 ==
 +
 +
We can prime factorize the solutions:
 +
A = 2 * 3 * 5
 +
B = 2 * 3 * 5 * 7 * 11 * 13
 +
C = 2 * 5 * 7
 +
D = 2 * 5 * 11 * 13
 +
E = 7 * 11 * 13
 +
 +
We can immediately eliminated B, D, and E since 13 only appears in 13!, 14!, 15, 16!, so 13^4 is a perfect square.
 +
Next, we can test if 7 is possible (and if it is not we can use process of elimination)
 +
7 appears in 7! to 16!
 +
14 appears in 14! to 16!
 +
So, there is an odd amount of 7's. Since 30 is not a divisor of 7, our answer is 70.
 +
 +
~aleyang

Revision as of 17:37, 15 November 2023

Problem

What is the least positive integer $m$ such that $m\cdot2!\cdot3!\cdot4!\cdot5!...16!$ is a perfect square?

Solution 1

Consider 2, there are odd number of 2's in $2!\cdot3!\cdot4!\cdot5!...16!$ (We're not counting 3 2's in 8, 2 3's in 9, etc).

There are even number of 3's in $2!\cdot3!\cdot4!\cdot5!...16!$ ...

So, original expression reduce to \begin{align*} m \cdot 2 \cdot 4 \cdot 6 \cdot 8 \cdot 10 \cdot 12 \cdot 14 \cdot 16 &\equiv m \cdot 2^8 \cdot (1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8)\\ &\equiv m \cdot 2 \cdot 3 \cdot (2 \cdot 2) \cdot 5 \cdot (2 \cdot 3) \cdot 7 \cdot (2 \cdot 2 \cdot 2)\\ &\equiv m \cdot 2 \cdot 5 \cdot 7\\ m &= 2 \cdot 5 \cdot 7 = 70 \end{align*}


Solution 2

We can prime factorize the solutions: A = 2 * 3 * 5 B = 2 * 3 * 5 * 7 * 11 * 13 C = 2 * 5 * 7 D = 2 * 5 * 11 * 13 E = 7 * 11 * 13

We can immediately eliminated B, D, and E since 13 only appears in 13!, 14!, 15, 16!, so 13^4 is a perfect square. Next, we can test if 7 is possible (and if it is not we can use process of elimination) 7 appears in 7! to 16! 14 appears in 14! to 16! So, there is an odd amount of 7's. Since 30 is not a divisor of 7, our answer is 70.

~aleyang

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2023_AMC_10B_Problems/Problem_15&oldid=203273"