# Chaitanya's Random Pages

## December 29, 2012

### A curious fact involving factors

Filed under: mathematics — ckrao @ 10:07 pm

Here is one of my favourite elementary number theory results.

If $a, b, c, d$ are positive integers such that $ab = cd$, then $a + b + c + d$ is composite.

I like it because it is not immediately obvious that the relationship $ab = cd$ should be related to the primality of the sum of these four numbers. As an example $12 = 2 \times 6 = 3 \times 4$ and $2 + 6 + 3 + 4 = 15 = 5 \times 3$.

To prove this, let $g = gcd(a,c)$. Then we may write $a = a_1 g$ and $c = c_1 g$ where $a_1$ and $c_1$ have no common factors. In other words, $g$ extracts the common factors from $a$ and $c$. This leads to $ga_1 b = g c_1 d$ or $a_1 b = c_1 d$. Since $a_1$ and $c_1$ have no common factors, this means all the non-unit factors of $a_1$ divide $d$ (and all the non-unit factors of $c_1$ divide $b$). Then we may write $d = a_1 h$ where $h$ is a positive integer so that $b = c_1 d / a_1 = c_1 h$.

Then

$\displaystyle a + b + c + d = a_1 g + c_1 h + c_1 g + a_1 h = (a_1 + c_1)(g + h)$

which is composite.

Note that we can similarly prove that $a^k + b^k + c^k + d^k$ is composite where $k$ is any positive integer.