# Characterization of the gcd

Proposition: (2.29, page 34) Suppose $$b\not=0$$. An integer $$d$$ is the greatest common divisor of $$a$$ and $$b$$ if and only if

• $$d\ge 0$$
• $$d$$ is a common divisor of $$a$$ and $$b$$
• If $$r$$ is a common divisor of $$a$$ and $$b$$, then $$r|d$$.

Proof: First suppose that these three conditions are true. Then $$d$$ is a common divisor, and by Proposition 2.1 (iv), if $$r$$ is any other common divisor of $$a$$ and $$b$$, then $$r|d$$ so $$|r|\le d$$. So $$d$$ is the greatest common divisor.

Now suppose $$d$$ is the greatest common divisor of $$a$$ and $$b$$. Then $$d\ge 0$$ and $$d$$ is a common divisor, so we just need to check the third condition. By the extended euclidean algorithm there are $$x$$ and $$y$$ so that $$ax+by=d$$. By Proposition 2.1 (ii), any common divisor of $$a$$ and $$b$$ divides $$ax+by=d$$, as we wanted to show.

# Least common multiple

Definition: A common multiple of two integers $$a$$ and $$b$$, with $$b\not=0$$, is any integer $$m$$ such that $$a|m$$ and $$b|m$$. The least common multiple of $$a$$ and $$b$$ is the smallest positive integer which is a common multiple of $$a$$ and $$b$$.

Theorem: The lcm of $$a$$ and $$b$$ is |ab/g| where $$g$$ is the gcd of $$a$$ and $$b$$.

Proof: We can assume $$a$$ and $$b$$ are non-negative as this does not affect the lcm. Because $$g$$ divides both $$a$$ and $$b$$, we have $$ab/g=a(b/g)=b(a/g)$$ so $$ab/g$$ is an integer and it is a common multiple of $$a$$ and $$b$$. Now let $$t$$ be any common multiple of $$a$$ and $$b$$. Find $$x$$ and $$y$$ so that $$ax+by=g$$. Then $$tax+tby=tg$$. Since $$t$$ is a common multiple of $$a$$ and $$b$$, we have $$tax$$ and $$tby$$ are both multiples of $$ab$$. So $$tax+tby=abs$$ for some integer $$s$$. We conclude that $$t=(ab/g)s$$, so that $$t$$ is a multiple of $$ab/g$$. This means $$t\ge (ab/g)$$ so $$ab/g$$ must be the least common multiple.

# Linear Diophantine Equations

A diophantine equation is an equation where the variables are restricted to integer values.

A linear diophantine equation in one variable is of the form $ax=b$ where $$a$$ and $$b$$ are integers and we want $$x$$ to be an integer. Clearly this has a solution exactly when $$a|b$$.

# Linear Diophantine Equations in 2 variables

A linear diophantine equation in two variables is an equation of the form $ax+by=c$ where $$a$$, $$b$$, and $$c$$ are integers.
Solving such an equation means finding integers $$x$$ and $$y$$ that satisfy the condition.

# Theorem on Linear Diophantine Equations

Theorem:

• The linear diophantine equation $$ax+by=c$$ has a solution if and only if $$\mathrm{gcd}(a,b)|c$$.

• If $$x_0$$, $$y_0$$ is one solution to the equation, and $$x$$ and $$y$$ is any other solution, then there exists an integer $$n$$ so that $x=x_0+n\frac{b}{d}\mathrm{\ \ and\ \ }y=y_0-n\frac{a}{d}$

# Proof of Main Theorem on Linear Diophantine Equations

1. If $$ax+by=c$$ has a solution, then $$\mathrm{gcd}(a,b)$$ must divide $$c$$. (This is Proposition 2.1 (ii))\$.
2. If $$\mathrm{gcd}(a,b)$$ divides $$c$$, then there are $$x$$ and $$y$$ such that $$ax+by=c$$. To find such $$x$$ and $$y$$, write $$c=\mathrm{gcd}(a,b)n$$. Use Euclid’s algorithm to find $$x$$ and $$y$$ with $$ax+by=\mathrm{gcd}(a,b)$$. Then $$anx+bny=n\mathrm{gcd}(a,b)=c$$. So $$nx$$ and $$ny$$ are a solution to the original equation.

# Proof continued

1. If $$(x,y)$$ and $$(x',y')$$ are two solutions to $$ax+by=c$$, then $a(x-x')+b(y-y')=0\mathrm{\ so\ } a(x-x')=b(y'-y).$ Divide both sides of this equation by $$d=\mathrm{gcd}(a,b)$$ to get $\begin{equation} \frac{a}{d}(x-x')=\frac{b}{d}(y-y') \end{equation}$ Remember that $$\mathrm{gcd}(a/d,b/d)=1$$. (This is Proposition 2.27 (ii)) At the same time, $$a/d$$ divides the left side of this equality, so it must divide the right side. By Proposition 2.28, this means that $$a/d$$ divides $$y-y'$$ so $$y-y'=(a/d)m$$ for some integer $$m$$. Also, $$b/d$$ divides $$x-x'$$ so $$x-x'=(b/d)m'$$. Therefore $\frac{a}{d}\frac{b}{d}m'=\frac{a}{d}\frac{b}{d}m$ so $$m=m'$$. In other words, $$x'=x-\frac{b}{d}m$$ and $$y'=y+\frac{a}{d}m$$ for some $$m\in\mathbb{Z}$$.
2. So far we know that any two solutions are related like $$(x,y)$$ and $$(x',y')$$ for SOME $$m$$. But in fact any $$m$$ works because $a(x-\frac{b}{d}m)+b(y+\frac{a}{d}m)=ax+by-\frac{ab}{d}m+\frac{ab}{d}m=ax+by=c.$ This concludes the proof of the main theorem.