Melissa has three different positive integers. she adds their reciprocals together and gets a sum of 1. what is the product of her integers?

First let us assign the three positive integers to be x, y, and z.

From the given problem statement, we know that:

(1/x) + (1/y) + (1/z) = 1

Without loss of generality we can assume x < y < z.

We know that:

1 = (1/3) + (1/3) + (1/3)

Where x = y = z = 3 would be a solution

However this could not be true because x, y, and z must all be different integers. And x, y, and z cannot all be 3 or bigger than 3 because the sum would then be less than 1. So let us say that x is a denominator that is less than 3. So x = 2, and we have:

(1/2) + (1/y) + (1/z) = 1

Therefore

(1/y) + (1/z) = 1/2

We also know that:

(1/4) + (1/4) = (1/2)

and y = z = 4 would be a solution, however this is also not true because y and z must also be different. And y and z cannot be larger than 4, so y=3, therefore

(1/2) + (1/3) + (1/z) = 1

Now we are left by 1 variable so we calculate for z. Multiply both sides by 6z:

3z + 2z + 6 = 6z

z = 6

Therefore:

(1/2) + (1/3) + (1/6) = 1

so {x, y, z}={2,3,6}