Let's begin with recalling the definition of the geometric mean.
The geometric mean of two positive numbers aand bis the numberxsuch thatx=sqrt(a* b).
To determine if the geometric mean for two positive integers is another integer, we will consider two cases.
The product of a and b is a perfect square.
The product of a and b is not a perfect square.
We will start with the first case. Let's choose a pair of positive integers that when multiplied give a perfect square. These numbers can be for example 2 and 8.
We can assume that for two positive integers whose product is a perfect square, the geometric mean is another integer. Now, we will consider the second case and choose two numbers that when multiplied do not give a perfect square. Let's choose for example 2 and 3.
As we can see, this time the geometric mean is not an integer. Therefore, we can say that the geometric mean for two positive integers is sometimes another integer.
