Number Theory

theorem 1

Consider the equation of the natural numbers

$ab=cd$

Given any the values of $b$ and $c$, find the minimum value of $a$ and $d$ that satisfies the equation. Consider the expansion of $b$ and $c$

$b = b_1 g$ $c = c_1 g$

where $g$ is the greatest common denominator of $b$ and $c$.

$a b_1 g = c_1 g d$ $a b_1 = c_1 d$ $a = \frac{c_1 d}{b_1}$

$b_1$ and $c_1$ are defined such that they do not share any factors. Therefor, $d$ must be a multiple of $b_1$.

$d = k b_1$ $a = \frac{k c_1 b_1}{b_1} = k c_1$

We now have an equation for all possible solutions for $a$ and $d$. The smallest values obviously correspond to $k=1$, which we will write below.

$a = \frac{c}{\mathrm{gcd}(b,c)}$ $d = \frac{b}{\mathrm{gcd}(b,c)}$