If I have $ n $-positive integers, and I compute their sum and product, is there any different group of $ n $-positive integers that will have the same sum and product?
For example, if $ a,…,z $ denote 26 positive integers, and we define:
\begin{align}
a+b+c+d+….+z &= \text{Sum} \\
a \cdot b \cdot c \cdot d \cdot …. \cdot z &= \text{Product}
\end{align}
Is there any way I can get the same Sum and Product from a different group of 26 (in this example) positive integers?
EDIT:
A friend of mine pointed out that knowing that we have a group of 3 that works, we can show that it works for all positive groups of $ n $ integers.
For Example:
$ \{3,3,10 \} $ and $ \{2,5,9 \} $ both yield Sum $=16 $ and Product $=90 $.
Now we can just continually add a number (let's say 1) as the next integer to get multiple solutions for $ n =4,5,6,… $.
Explicitly, $ \{ 3,3,10,1 \} $ and $ \{ 2,5,9,1 \} $ both give Sum$=17$ and Product$=90$.
Best Answer
Given any of lots of pairs $\{a,b\}, \{d,e\}$, you can add a number to each to make matching sum and product. We want $$a+b+c=d+e+f\\abc=def\\f-c=a+b-d-e\\\frac fc = \frac {ab}{de}$$ and we can solve the last two to get $$c=(a+b-d-e)\frac 1{\frac {ab}{de}-1}\\f=(d+e-a-b)\frac 1{\frac {de}{ab}-1}$$ As long as $a+b \neq d+e$ and none are zero we have a solution. As long as the pair with the greater sum also has the greater product both $c,f$ will be positive.