Visualizing Greatest Common Divisor (gcd)

Question

If a divides b and leaves a remainder c, then gcd (a,b)= gcd (a,c).

The book has given a detailed proof for it. But I want to understand it intuitively.

On diving 15 with 4, remainder is 3. And gcd (15,4) = gcd (3,4). But why is it happening?

Answer 1

$b$ is divisible by $a$ leaving remainder as $c$ hence we can write

$b = ak+c$ where $k$ is an integer and $k\geqslant0$

Now let's say $gcd(a,b)=x$

$\Rightarrow gcd(a,ak+c)=x$

$\Rightarrow a=mx$ and $c=nx$ such that $gcd(m,mk+n)=1$ i.e. $m$ and $mk+n$ don't have any common divisor. (Then only $x$ will be gcd else a multiple of $x$ will become gcd).

Taking Right Hand Side- Now we have to prove that $gcd(mx, nx)=x$ i.e. we have to prove that $m, n$ are coprime (i.e. $m$ and $n$ don't have any common factor).

We already know that $m, mk+n$ don't have any common factor so we can easily deduce that $m$ and $n$ also don't have any common factor. (Proof by contradiction- Let's say $m$ and $n$ have a common factor then $mk+n$ will be divisible by $m$ which is impossible so $m$ and $n$ can't have a common factor.)

Hence it is proved that $m$ and $n$ are coprime so $gcd(mx, nx)$ will be $x$ because $gcd(m,n)=1$