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Showing posts from August, 2014

### Learn One Weird Trick And Easily Solve The Product-Sum Problem

A tribute to Martin Gardner.

For which sets of positive integers does the sum equal the product? For example, when does $$x_1 + x_2 = x_1\cdot x_2$$? It's easy to see that this is only true when $$x_1 = x_2 = 2$$.

In the general case our equality is $$\sum_{i=1}^{n} x_i= \prod_{i=1}^{n} x_i$$. We can rearrange terms to give $x_1+x_2+\sum_{i=3}^{n} x_i= x_1\cdot x_2\cdot \prod_{i=3}^{n} x_i,$ and this in turn factors nicely to give us $\left( x_1\cdot \prod_{x=3}^{n} x_i - 1\right)\cdot \left( x_2\cdot \prod_{x=3}^{n} x_i - 1\right) = \left( \prod_{x=3}^{n} x_i \right)\cdot \left(\sum_{x=3}^{n} x_i \right) + 1.$ How does this help? Consider the case $$n=5$$, and without loss of generality assume $$x_i \ge x_{i+1}$$ for all $$i$$. If $$x_3=x_4=x_5=1$$ our factorized equation becomes $(x_1-1)\cdot(x_2-1)=4,$ with the obvious solutions $$x_1=5, x_2=2; x_1=3, x_2=3$$. The only remaining case to consider is $$x_3=2$$, as any other case forces  \( \sum_{i=1}^{n} x_i < \prod_{…