2013 UNCO Math Contest II Problems/Problem 2
A number is equal to . What is the smallest positive integer such that the product is a perfect cube?
We can factor into . There are already two factors of two, so we only need to multiply it by to get two factors of three, giving us .
To find the perfect cube, we need all of the prime factors to be to the third power. Because is squared, we need to multiply by a power of , giving us , which is . Because we only have one power of three, we need two more, so we multiply , giving us , which is a perfect cube.
To find a perfect 6th power, we multiply by to get . We know that the factorization of this number is . This means that this number is . We need a perfect 6th, so we multiply by to get , which is , or .
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