## Ask Professor Puzzler

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Hi Professor, I saw this riddle on a website: "The sum of three numbers is equal to their product. What are the numbers?" They gave an answer of "1,2, and 3". I wondered if this is the only answer. verity

Hi Verity,

In order to answer this question, we should start by writing the problem algebraically:

abc = a + b + c

What you should notice right away is that there are three unknowns (three variables) and only one equation. This makes it seem very likely that there will be multiple solutions. The first thing I tried with this equation was to solve for one variable. Shall we solve for a? Here are my steps:

abc = a + b + c

abc - a = b + c (subtract a from both sides)

a(bc - 1) = b + c (factor out an a)

a = ^{(b + c)}/_{(bc - 1)} (divide both sides by bc - 1)

You know what this looks like to me? It looks like we can pick any values we want for b and c, as long as b and c aren't reciprocals of each other, and we'll get a value for a. b and c can't be reciprocals of each other because then (bc - 1) would be zero, and we'd have a division by zero, which is illegal, immoral and socially unacceptable.

So let's pick a couple values. b = 3; c = 4.

a = ^{(3 + 4)}/_{(12 -1 ) }= ^{7}/_{11}.

Let's test our answer:

3 + 4 + ^{7}/_{11} = ^{84}/_{11}.

3(4)(^{7}/_{11}) = ^{84}/_{11}.

Sure enough, they're the same! So the answer to your question is, no, their solution is not the only one; there are an infinite number of solutions.

I suspect that the site intended to make it more challenging by requiring you to use only integers, or maybe distinct integers. But that doesn't change the fact that there's an infinite number of solutions. For any integer x, the following three numbers work:

x, 0, -x