2009 AMC 8 Problems/Problem 17
Contents
Problem
The positive integers and are the two smallest positive integers for which the product of and is a square and the product of and is a cube. What is the sum of and ?
Video Solution by OmegaLearn
https://youtu.be/7an5wU9Q5hk?t=2768
Video Solution
https://www.youtube.com/watch?v=ZuSJdf1zWYw
Solution
The prime factorization of . If a number is a perfect square, all of the exponents in its prime factorization must be even. Thus we need to multiply by a 2 and a 5, for a product of 10, which is the minimum possible value of x. Similarly, y can be found by making all the exponents divisible by 3, so the minimum possible value of is . Thus, our answer is .
Solution 2 (Using Answer Choices)
From the question's requirements, we can figure out is . Then we can use the answer choices to find what is. Let's start with . If was right, then . We can multiply by and get , which isn't a perfect cube. Then we move to . , so if is right. Then we multiply by to get , which is . Therefore, our answer is because and .
~Trex226
See Also
2009 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 16 |
Followed by Problem 18 | |
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