Answer:
Step-by-step explanation:
Given 3 positive integers x, y and z, if their product is 125, then;
xyz = 125 .......... 1
Sum of the integers S(x, y, z) = x+y+z ........... 2
From equation 1; x = 125/yz ........3
Substitute equation 3 into 2
S(x, y, z) = 125/yz+y+z
Sy = -125/y²z + 1
Sz = -125/z²y + 1
For the sum to be minimum, Sy = Sz = 0
-125/y²z + 1 = 0
-125/y²z = -1
-125 = -y²z
125 = y²z
Also; is Sz = 0
-125/z²y + 1 = 0
-125/z²y = -1
-125 = -z²y
125 = z²y
Hence y²z = z²y = 125
From y²z = 125, z = 125/y²
Substitute z = 125/y² into the equation 125 = z²y
125 = (125/y²)²× y
125 = 125²/y⁴ × y
125 = 125²/y³
1 = 125/y³
y³ = 125.
y = 5
Since 125 = y²z
5²z = 125
25z = 125
z = 5
From the product xyz = 125
x(5)(5) = 125
25x = 125.
x = 5
Hence x = y = z = 5
