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# help and explain :)

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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