Can someone help me understand this statement:
If $ax+by=d$ then $\gcd(a,b)$ divides $d$.
Bezout's identity states that:
the greatest common divisor $d$ is the smallest positive integer that can be written as $ax + by$
However the definition of $\gcd(a, b)$ is the largest positive integer which divides both $a$ and $b$.
I'm am completely lost. If anyone could provide some sort of layout to help me sort this out I would be really happy.
Take a common divisor $k$ of $a$ and $b$. That means that we can write $a=ka'$, $b=kb'$. Then, if $d=ax+by$, $$d=a'kx+b'ky=k(a'x+b'y)$$ so we see that $k$ also divides $d$. Since $\gcd(a,b)$ is a common divisor of $a$ and $b$ (namely, the greatest), it also holds the property. Actually, the idea behind this is very simple: the sum of two multiples of the same number is also a multiple of the same number.
Note that $\gcd(a,b)$ is, hence, the least positive integer that can be written as $ax+by$, because every number of the form $ax+by$ must be a multiple of any common divisor of $a$ and $b$, and in particular, must be a multiple of $\gcd(a,b)$ so $ax+by$ can't be smaller than it (being still positive, of course).
Example: let $a=60$, $b=75$. The gcd is $15$, so $ax+by=60x+75y$ must be a multiple of $15$, because it is the sum of two multiples of $15$.
I hope this helps a bit.
Hint: For three numbers $a,b,c$, if $c$ divides $a$ and $c$ divides $b$, then $c$ divides $a+b$.
I'd like to offer a perspective from algebra that made it click for me. Consider John B. Fraleigh's definition from 'Introduction to Abstract Algebra':
Let $r$ and $s$ be two positive integers. The positive generator $d$ of the cyclic group $H = \{nr + ms \mid m,n \in \mathbb{Z}\}$ under addition is the $\textbf{greatest common divisor}$ of $r$ and $s$.
Let's consider a numeric example, same as above: $r = 60$ and $s = 75$. Their gcd is obviously $d = 15$ and all the multiples of $15$ are in $H$: $(k)(75)+(-k)(60) = 15k \in H, k \in \mathbb{N}$. Let's show that there are no integers smaller than $d$ in $H$.
Suppose to the contrary that there exists an integer $c \in \mathbb{N}$ such that $c < d$ and $c \in H$. Then $c = nr + ms$. Since $d$ divides $r$ and $s$, we can write $r = di$ and $s = dj$ for some $i,j \in \mathbb{N}$. Now $c = ndi + mdj$, which means $\frac{c}{d} = ni + mj \in \mathbb{Z}$, which is a contradiction as our assumption meant $0<\frac{c}{d} < 1$.
So there are no positive integers smaller than $d \in H$. Since $d$ divides every number of the form $nr + ms$ and is the smallest positive element, it generates the set.
