Find three positive integers x, y, and z that satisfy the given conditions. The product is 64 and the sum is a minimum. (x, y, z) = ( )

To find three positive integers x , y , and z that multiply together to give 64 and have the smallest possible sum, we need to break down the problem into a few steps.
First, let's find the factor combinations of 64. The number 64 is a power of 2, specifically 64 = 2 6 . This means there are several combinations of integers that multiply to 64, such as:

( 1 , 1 , 64 )

( 1 , 2 , 32 )

( 1 , 4 , 16 )

( 1 , 8 , 8 )

( 2 , 4 , 8 )


Next, for each combination, we calculate the sum x + y + z :

1 + 1 + 64 = 66

1 + 2 + 32 = 35

1 + 4 + 16 = 21

1 + 8 + 8 = 17

2 + 4 + 8 = 14


Among these, the combination ( 2 , 4 , 8 ) gives the smallest sum, which is 14.
Therefore, the integers x , y , and z are 2 , 4 , and 8 , respectively. This set of integers has the product 2 × 4 × 8 = 64 and the minimum sum of 2 + 4 + 8 = 14 .

Answered by RyanHarmon181 | 2025-07-21