Perfect Square

What is the smallest positive integer n such that 16!/n is a perfect square?
Source: NCTM Mathematics Teacher, December 2005

SOLUTION
16!=16\cdot 15\cdot 14\cdots 3\cdot 2\cdot 1
16=2^4
15=3\cdot 5
14=2\cdot 7
13=1\cdot 13
12=2^2\cdot 3
11=1\cdot 11
10=2\cdot 5
9=3^2
8=2^3
7=1\cdot 7
6=2\cdot 3
5=1\cdot 5
4=2^2
3=1\cdot 3
2=1\cdot 2
16!=2^{15}\cdot 3^6\cdot 5^3\cdot 7^2\cdot 11\cdot 13
=(2^7)^2\cdot 2\cdot (3^3)^2\cdot 5^2\cdot 5\cdot 7^2\cdot 11\cdot 13
n will be smallest when we make the numerator as large a perfect square as it can be.
n=2\cdot 5\cdot 11\cdot 13=1430

Answer: 1430

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About mvtrinh

Retired high school math teacher.
This entry was posted in Problem solving and tagged , , , , , , , , . Bookmark the permalink.

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