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Let x be the smallest positive integer such that 1584 * x  is a perfect cube, and let y be the smallest positive integer such that xy is a multiple of 1584. Compute y

 Jul 6, 2021
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First let's take the prime factorization of 1584,

\(2^4\cdot 3^2\cdot11 = 1584\)

To make \(1584\cdot x \), each of the exponents in the prime factoriziation of the cube need to be a multiple of 3. To make that, the prime factoriziation of x needs to create that by having the correct exponents of,

\(2^2 \cdot 3 \cdot 11^2 = 1452\)

Making the cube's prime factorization,

\(2^6 \cdot 3^3 \cdot 11^3 = 2299968\)

\(\sqrt[3]{2299968} = 132\)

So our cube is actually a cube. Now that we know that x = 1452, we can use LCM to find a common multiple of 1452 and 1584 to find y. 

The LCM of 1452 and 1584 is 17424.

\(\frac{17424}{1452} = 12\)

So y = 12

 

Sorry if my method was kind of confusing lol

 Jul 6, 2021

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