Question

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

Answer 1

If the three integers are
`a,b,c`, then we have

`\frac1a+\frac1b+\frac1c=1`

We can combine the fractions on the left side:

`(bc)/(abc)+(ac)/(abc)+(ab)/(abc)=(bc+ac+ab)/(abc)=1`

`\implies abc=bc+ac+ab`

Answer 2

36

Explanation:

Let the three positive integers be x, y, and z. Then

1/x + 1/y + 1/z = 1

Assume x = 2.

Then 1/x = ½ and 1/y + 1/x = 1/2

Divide the second portion (1/y + 1/z) into three parts.

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

Combine two of the fractions.

1/2 = 1/6 + 2/6

1/2 = 1/6 + 1/3

1/2 + 1/3 + 1/6 = 1

The integers are 2, 3, and 6.

2 × 3 × 6 = 36

The product of Melissa’s integers is 36.