X is the product of four positive integers. One of the four numbers, 58 was replaced by 85 and the new product was found to be Y. If the difference between X and Y is 13,608, find the minimum sum of the other three numbers.
Question
X is the product of four positive integers. One of the four numbers, 58 was replaced by 85 and the new product was found to be Y. If the difference between X and Y is 13,608, find the minimum sum of the other three numbers.
Solution
Let's denote the three other numbers as a, b, and c. So, we have:
X = 58 * a * b * c Y = 85 * a * b * c
We know that Y - X = 13,608. Substituting the expressions for X and Y from above, we get:
85 * a * b * c - 58 * a * b * c = 13,608 27 * a * b * c = 13,608
So, a * b * c = 13,608 / 27 = 504.
We want to minimize the sum a + b + c. The minimum sum of three numbers given their product is achieved when the three numbers are as close to each other as possible.
Since 504 = 8 * 7 * 9, the minimum sum a + b + c is 8 + 7 + 9 = 24.
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