Find five positive integers whose reciprocals sum to 1

One can first try to compose the sum of 1 with smaller number of reciprocals. Let's start with smallest positive integers. 1/2 + 1/3 < 1, but 1/2 + 1/3 + 1/4 > 1, and the part 1 - (1/2 + 1/3) is actually 1/6. So, with 3 positive integers we can have 1/2 + 1/3 + 1/6 = 1.

Now we can try to replace 1/6 with smaller reciprocals, and we need exactly 3 of them, so the problem sort of repeats itself - now we need to find 3 such integers, that their reciprocals sum to 1/6. This is the same problem as we just solved, only scaled 1/6 times, so instead of 1/2, 1/3 and 1/6 we can have them divided by 6 - so, 1/12, 1/18 and 1/36.

So now we have five numbers - 1/2, 1/3, 1/12, 1/18 and 1/36 - summing to 1 and their reciprocals are integers.

You suspect that's not the only solution - after all, original 1/2 + 1/3... was selected rather arbitrarily, as the largest numbers with reciprocals still less than 1.

I came at a solution doing the following:

Well, I know this:

   1 = 1/2 + 1/2
   

We're trying to get to 5 terms. So let's split 1/2 into two terms.

   1 = 1/2 + 1/4 + 1/4
   

Hmm, let's do that again - this time splitting 1/4.

   1 = 1/2 + 1/4 + 1/8 + 1/8
   

Ok, I'm pretty close. Can I split 1/8 into 1/9 and some mystery term?

   1 = 1/2 + 1/4 + 1/8 + 1/9 + 1/N
   

Well, that's not hard to solve. N = 72.

1 = 1/2 + 1/4 + 1/8 + 1/9 + 1/72