Let's denote the two positive integers as A and B. We are given that the greatest common divisor (GCD) of A and B is (x + 5), and their least common multiple (LCM) is x(x + 5).
We know that for any two positive integers A and B, the relationship between their GCD and LCM is given by:
GCD(A, B) * LCM(A, B) = A * B
Substituting the given values, we get:
(x + 5) * x(x + 5) = 50 * B
Simplifying the equation:
x(x + 5)^2 = 50 * B
Since x and (x + 5) are both positive integers, we can consider the smallest possible value for x. Let's start with x = 1:
1(1 + 5)^2 = 6^2 = 36
We see that 36 is a factor of 50 * B. The smallest possible value of B can be found by dividing 50 by the GCD (x + 5), which is 6:
B = 50 / (x + 5) = 50 / 6 = 25/3
Since B must be a positive integer, the smallest possible value of B is 25. Therefore, the other positive integer is 25.