The sum of the reciprocals of 3 natural numbers is 1. Find all such triplets.The first is {3, 3, 3}?

Question

Name two more triplets that work.....Example: 1/3 + 1/3 + 1/3 = 1
{3, 3, 3} .......First Triplet
? ..........Second Triplet
?...........Third Triplet

Answer 1

Here you go:
{3, 3, 3} = 1/3 + 1/3 + 1/3
{2, 4, 4} = 1/2 + 1/4 + 1/4 *
{2, 3, 6} = 1/2 + 1/3 + 1/6

* You're welcome, answerer #1; good job copying my answer :)

Answer 2

Others have shown the solution. I can provide the proof that these are the only solution and there is no other solution.

it is sum of 3 numbers so largest number cannot be < 1/3. if this is so then sum < 1

so largest number is 1/3 or 1/2.
In case 1/3
in this case other numbers have to be 1/3 and 1/3 . as sum of the other two has to be 2/3 so 2nd largest cannot any thing other than 3
so reciprocal of 3,3,3.

in case 1/2

the sum of 2 other numbers = 1/2 so 2nd largest cannot be < 1/4

2nd largest cannot be 1/2 because then 3rd largest 0
2nd largest if 1/3 the 3rd number = 1/6 so set is (2,3,6)
2nd largest if 1/4 then 3rd number = 1/4 so (2,4,4,)

there cannot be any other solution(all are enumerated)

Answer 3

The second is 1/2, 1/3, 1/6. Hmm, you say there is one more...?

Oh yeh, 1/2, 1/4, 1/4

Answer 4

do they have to be the same number? if not:
2,4,4