Question

_{If $a>o$and $b>0$, prove that $\left[a,b\right]=\frac{ab}{\left(a,b\right)}$.}

Short answer

It is proved that$\left[a,b\right]=\frac{ab}{\left(a,b\right)}$

Step-by-step solution

Consider the given situation

Consider the two positive integers a and b such that $a>0$ and $b>0$. We need to prove that $\left[a,b\right]=\frac{ab}{\left(a,b\right)}$.

Assume that $d=\left(a,b\right)$.Then, we can write that$a=du$ and$b=dv$ for some integers u and v.

Write the common multiple as follows:

role="math" localid="1646137527994" $\begin{array}{rcl}m& =& \frac{ab}{d}\\ & =& \frac{\left(du\right)\left(dv\right)}{d}\\ & =& \frac{d^{2}uv}{d}\\ & =& d\left(uv\right)\end{array}$

Proove that a,b=aba,b

$\left[a,b\right]=\frac{ab}{\left(a,b\right)}$As the positive integer has a common multiple m, integer c is the other least common multiple of integers a and b.

So, the least common multiple is

.$\begin{array}{rcl}d\left(uv\right)& =& \frac{ab}{d}\\ & =& \frac{ab}{\left(a,b\right)}\end{array}$

Hence, the common multiple is always less than or equal to the other multiple. This implies that

$m=d\left(uv\right)$

$\left[a,b\right]=\frac{ab}{\left(a,b\right)}$

It is proved that$\left[a,b\right]=\frac{ab}{\left(a,b\right)}$