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Ref to the question in Unusual 5th grade problem, how to solve it.

Find a positive integer solution $(x,y,z,a,b)$ for which

$$\frac{1}{x}+ \frac{1}{y} + \frac{1}{z} + \frac{1}{a} + \frac{1}{b} = 1\;.$$

Here is my question: How many solutions for this question?

There was a C++ program for the answer under 100. But you can easily find answer outside 100, as [x:5, y:220, z:4, a:2, b:22].

I can find 2477 answers for number <= 1806. When we change the upper limit, are there more answers? Is the total number of solutions limited? I guess so, but not able to prove it. Can you prove it?

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The total number of solutions is limited. In fact, the largest possible value for the variables would arise from the greedy algorithm (choosing the largest possible $\frac{1}{n}$ each time), which gives

$$ 1 =\frac{1}{2} + \frac{1}{3} + \frac{1}{7} + \frac{1}{43} + \frac {1}{ 1806}$$

Hence, you're done (assuming that your calculations are correct).

As to how to show that the greedy algorithm yields the largest possible value, read up on the Sylvester Sequence.

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