Question

If $a|c,b|c$and$\left(a,b\right)=d$ , prove that$ab|cd$.

Short answer

It is proved that if $a|c,b|c$and role="math" localid="1646046151630" $\left(a,b\right)=d$, then$ab|cd$.

Step-by-step solution

1. Definition of Greatest Common Divisor:

Let be the largest integer that divides the integers . Then the integer can be designated as:

$d=\left(a,b\right)$

Where$a,b\ne 0$

2. Proof using Definition and Theorem:

Let $d=\left(a,b\right)$.. Then, we get $d|a$and $d|b$.

Therefore, we have:

$a=d{a}_1.......\left\{a_1\in Z\right\}$

$b=d{b}_1.......\left\{b_1\in Z\right\}$

Since $a|c$, therefore,

$c=au=d{a}_{1}u........\left\{u\in Z\right\}$

Similarly, we have:

$c=bv=d{b}_{1}v........\left\{v\in Z\right\}$

Now, $a_{1}u=c|d=b_{1}v$; this implies that:

$a_{1}v\Rightarrow v=a_{1}w.......\left\{w\in Z\right\}$

So, we have:

$c=d{a}_1{b}_{1}w$

$cd=d^2{a}_1{b}_{1}w$

$cd=abw$

3. 

Thus,$ab|cd$

Hence, it is proved that if $a|c,b|c$and $\left(a,b\right)=d$, then $ab|cd$.