Question

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

Answer:

36

Step-by-step 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.

Answer 2

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

$\dfrac1a+\dfrac1b+\dfrac1c=1$

We can combine the fractions on the left side:

$\dfrac{bc}{abc}+\dfrac{ac}{abc}+\dfrac{ab}{abc}=\dfrac{bc+ac+ab}{abc}=1$

$\implies abc=bc+ac+ab$