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x,y, and z are positive integers such that gcd(x, y, z) = 14 and lcm(x, y, z) = 630 and y = 3x. What is the smallest possible value of x + y + z?

 Jul 12, 2022
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x * y * z = 630 x 14

3x^2 * z = 630 * 14

x^2 * z = 210 * 14

x^2 * z = 2^2 * 3 * 5 * 7^2

 

In order to minimize x + y + z, we get:

x = 7, z = 60... Therefore the minimum value of x + y + z = 88.

 Jul 13, 2022

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