Question

There are three different positive integers whose reciprocals add up to 1. What is the product of those three integers?

Answer 1

Tried anything?

Answer 2

Reciprocals means this:
5/4 reciprocal is 4/5

Answer 3

well i mean at first i thought the three numbers were 3. But then i saw that they were all supposed to be different

Answer 4

and @aaron i know what a reciprocal is

Answer 5

As a start, setup an equation

Answer 6

1/z+1/x+1/x=1
x*x*x=?

Answer 7

Let the 3 number be x,y,z respectively.

So,
\[\frac{ 1 }{ x } + \frac{ 1 }{ y } + \frac{ 1 }{ z }= 1\]

Answer 8

Take the LCM
\[\frac{ yz }{ xyz } + \frac{ xz }{ xyz } + \frac{ xy }{ xyz } =1\]

Answer 9

yeah that's where i'm stuck

Answer 10

product xyz = yz+xz+xy

Answer 11

the only saving grace is that they're integers

Answer 12

xyz = yz+xz+xy

Now we have to use the try and error method
we have substitute x , y and z with number such that the sum of their reciprocal is 1.

Answer 13

I think I found the 3 numbers
there 2,3,and 6

1/2 + 1/3 + 1/6
LCM = 6
3/6 + 2/6 + 1/6
3+2+1 = 6
and 6/6 = 1

Answer 14

looks good, but are there any others?

Answer 15

is it 2,3, and 6? because 1/2=3/6 and 1/3=2/6 and 2/6+3/6+1/6=6/6=1?

Answer 16

Yep! :)

Answer 17

Nice

Answer 18

thank you so much!