2023 AMC 10B Problems/Problem 15
Contents
Problem
What is the least positive integer such that is a perfect square?
Solution 1
Consider 2, there are odd number of 2's in (We're not counting 3 2's in 8, 2 3's in 9, etc).
There are even number of 3's in ...
So, original expression reduce to
Solution 2
We can prime factorize the solutions:
We can immediately eliminate B, D, and E since 13 only appears in , so 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 which is .
~aleyang
Solution 2
First, we note . Simplifying the whole sequence and cancelling out the squares, we get . Prime factoring and cancelling out the squares, the only numbers that remain are and . Since we need to make this a perfect square, . Multiplying this out, we get .
~yourmomisalosinggame (a.k.a. Aaron)