Thread: Help me to solve it, please.

1. Help me to solve it, please.

A unit fraction is a fraction of the form 1/n where n is a positive integer. Note that the unit unifraction 1/11 can be written as the sum of two unit fractions in the following three ways:
1/11=1/12+1/132=1/22+1/22=1/132+1/12
Are there any other ways of decomposing
1/11 into the sum of two unit fractions? In how many ways can we write 1/60 as the sum of two unit fractions? More generally, in how many ways can the unit fraction 1/n be written as the sum of two unit fractions?
In other words, how many ordered pairs (
a, b) of positive integers a, b are there for which 1/n=1/a+1/b?

2. Re: Help me to solve it, please.

For 1/60 = sum of 2 unit fractions, there are 23 ways:
1: 1/61 + 1/3660
2: 1/62 + 1/1860
3: 1/63 + 1/1260
...
21:1/108 + 1/135
22: 1/110 + 1/132
23: 1/120 + 1/120

If you insist that switching 'em around is another set, then add 22, for 45.

Nothing magic: used dumb code on a double looper....

1/60 = 1/a + 1/b
ab / (a + b) = 60

3. Re: Help me to solve it, please.

the number of solutions equals the number of positive divisors of $n^2$

Proof: $n^2=(a-n)(b-n)$

$n=60$, $60^2=2^4 3^2 5^2$ so $45$ solutions

$n=11$ three solutions