# Thread: Find all 4-tuples of distinct positive integers

1. ## Find all 4-tuples of distinct positive integers

Find all 4-tuples (a, b, c, d) of distinct positive integers so that a < b < c < d and 1/a + 1/b + 1/c + 1/d = 1.

My attempt:

If a = 1, 1/b + 1/c + 1/d = 0, which is impossible.

If a >= 3, b >= 4, c >= 5, d >= 6 implies that 1/a + 1/b + 1/c + 1/d < 1/3 + 1/4 + 1/5 + 1/6 < (20 + 15 + 12 + 10)/60 = 57/60 < 1.

So a = 2, 1/b + 1/c + 1/d = 1/2 and b >= 3.

How do I proceed?

2. ## Re: Find all 4-tuples of distinct positive integers

How you solved that?

3. ## Re: Find all 4-tuples of distinct positive integers

Originally Posted by Also sprach Zarathustra
How you solved that?
If a = 1, 1/b + 1/c + 1/d = 0, which is impossible.
If a >= 3, b >= 4, c >= 5, d >= 6 implies that 1/a + 1/b + 1/c + 1/d <= 1/3 + 1/4 + 1/5 + 1/6 = (20 + 15 + 12 + 10)/60 = 57/60 < 1.
So a = 2, 1/b + 1/c + 1/d = 1/2 and b >= 3.

If b >= 6, c >= 7, d >= 8, 1/b + 1/c + 1/d <= 1/6 + 1/7 + 1/8 = (28 + 24 + 21)/168 = 73/168 < 1/2 - Contradiction!
So b = 3, 4 or 5.

Case 1: If b = 3, 1/c + 1/d = 1/2 - 1/3 = 1/6, c >= 4

If c <= 6, 1/d = 1/6 - 1/c <= 0, which is impossible.
If c >= 11, d >= 12, 1/c + 1/d <= 1/11 + 1/12 = 23/242 - Contradiction!
So c = 7, 8, 9 or 10.

If c = 7,
1/7 + 1/d = 1/6
1/d = 1/42
d = 42
If c = 8,
1/8 + 1/d = 1/6
1/d = (4 - 3) / 24
d = 24
If c = 9,
1/9 + 1/d = 1/6
1/d = (3 - 2)/18
d = 18
If c = 10,
1/10 + 1/d = 1/6
1/d = (5 - 3)/30
d = 30/2 = 15

Case 2: If b = 4, 1/c + 1/d = 1/4, c >= 5.
If c >= 8, d >= 9, 1/c + 1/d <= 1/8 + 1/9 = 17/72 - Contradiction!
So c = 5, 6 or 7.

If c = 5,
1/5 + 1/d = 1/4
1/d = 1/20
d = 20
If c = 6,
1/6 + 1/d = 1/4
1/d = 1/12
d = 12
If c = 7,
1/7 + 1/d = 1/4
1/d = 3/28
d = 28/3, which is not a positive integer.

Case 3: If b = 5, 1/c + 1/d = 1/2 - 1/5 = 3/10, c >= 6.
If c >= 7, d >= 8, 1/c + 1/d <= 1/7+1/8 = 15/56 < 3/10 - Contradiction!
So c = 6.
1/6 + 1/d = 3/10
1/d = (9 - 5)/30 = 4/30 = 2/15
d = 15/2, which is not a positive integer.

Thus, there are 6 solutions: (2, 3, 7, 42), (2, 3, 8, 24), (2, 3, 9, 18), (2, 3, 10, 15), (2, 4, 5, 20) and (2, 4, 6, 12).