Question

Prove that$\left(a,b\right)=\left(a,b+at\right)$ for every $t\in \mathrm{\mathbb{Z}}$.

Short answer

It is proved that $\left(a,b\right)=\left(a,b+at\right)$for every $t\in \mathrm{\mathbb{Z}}$.

Step-by-step solution

Definition of Greatest Common Divisor: 

Let $d$ be the largest integer that divides the integers $a$and$b$. Then the integer $d$can be designated as:

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

Where, $a,b\ne 0$.

Proof using Linear Combination: 

Let $d$ be the greatest common factor such that:

$$\begin{gathered} d=\left(a,b\right) \\ e=\left(a,b+at\right) \end{gathered}$$

Here, $\left(b+at\right)$ is a linear combination of $a$and$b$. Then,

$d|\left(b+at\right)$

$d|e$......................$\left\{1\right\}$

Also, we have:

$b=a\left(-t\right)+\left(b+at\right)$

Here, $b$ is a linear combination of $a$and$\left(b+at\right)$. Then,

$e|d$...................$\left\{2\right\}$

From expressions 1 and 2, we have:

role="math" localid="1646284028704" $$\begin{gathered} d=e \\ or,\left(a,b\right)=\left(a,b+at\right) \end{gathered}$$

Hence, it is proved that $\left(a,b\right)=\left(a,b+at\right)$ for every $t\in \mathrm{\mathbb{Z}}$.