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# Number Theory

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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

Feb 1, 2022

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1584 = 2^4 * 3^2 * 11

To be a perfect cube, a number's prime factors' exponents must all be multiples of 3. So, the smallest value of x is:

$$\frac{2^6 \cdot 3^3 \cdot 11^3}{2^4 \cdot 3^2 \cdot 11} = \boxed{1452}$$

In other terms, x = 1452 = 2^2 * 3 * 11^2. The product xy must be a multiple of 1584, so,

xy = 2^2 * 3 * 11^2 * y = 2^4 * 3^2 * 11 * n

y must have 2^2 * 3 as a factor, so the smallest value of y is 12.

Feb 1, 2022