If $\gcd(a,b)=1$, then $\operatorname{lcm}(a,b)=ab$

Question

Could someone please tell me whether my proof is okay? I was inspired by this proof and yet I was confused what they meant by common divisor and how it connects to common multiple.

If $\gcd(a,b)=1$, then $\operatorname{lcm}(a,b)=ab$.

Assume $\gcd(a,b)=1$. Then $a$ and $b$ are relatively prime. Then let $a=p_{1}p_{2}\cdots p_{n}$ and $b=q_{1}q_{2}\cdots q_{m}$ for primes $p_{i}$ and $q_{j}$ with $p_{i}\neq q_{j}$ for all $i,j$. Let $m$ be a common multiple of $a$ and $b$. Then $a \mid m$ and $b \mid m$. Since $\gcd(a,b)=1$, $ab \mid m$. Then there exists a positive integer $k$ such that $m=abk=(p_{1}p_{2}\cdots p_{n})(q_{1}q_{2}\cdots q_{m})k$. Then the least common multiple of $a$ and $b$ would be when $k=1$. Then $\operatorname{lcm}(a,b)=ab$.

I really do not find my proof very good. I get confused in the last part, connecting $abk$ with $\operatorname{lcm}(a,b)$. Any help is appreciated…

Answer 1

Your proof is correct, but you need not use the decomposition of an integer into prime factors. More elementary arguments work here.

As you said, because $\operatorname{gcd}(a,b)=1$, any common multiple of $a$ and $b$ is in fact a multiple of $ab$. Conversely, any multiple of $ab$ is a common multiple of $a$ and $b$.

Hence, the least common multiple must be $ab$ itself.

NB: to justify that any common multiple of $a$ and $b$ is in fact a multiple of $ab$, we can proceed by using Bézout's theorem. We are given a relation $$au+bv=1$$ where $u,v$ are integers. Now, let $m$ be a common multiple of $a$ and $b$. Let us write $m=aa'=bb'$. Then $$m=aa'=(a^2u+abv)a'=(aa')au+(ab)a'v=(bb')au+(ab)a'v=ab(b'u+a'v)$$ With this, the proof is complete with only elementary arguments.

NB2: As @abc... stated, it is in general true that $ab=\operatorname{gcd}(a,b)\operatorname{lcm}(a,b)$. We can now prove this without using the decomposition into prime factors, thanks to the fact that if $k$ is any integer, then $k\operatorname{gcd}(a,b)=\operatorname{gcd}(ka,kb)$ and $k\operatorname{lcm}(a,b)=\operatorname{lcm}(ka,kb)$.

Indeed, given $a,b$, it follows from this property than $\operatorname{gcd}(\frac{a}{\operatorname{gcd}(a,b)},\frac{b}{\operatorname{gcd}(a,b)})=1$. Applying the proven result, we know that $\operatorname{lcm}(\frac{a}{\operatorname{gcd}(a,b)},\frac{b}{\operatorname{gcd}(a,b)})=\frac{ab}{\operatorname{gcd}(a,b)^2}$. Now, multiply both sides by $\operatorname{gcd}(a,b)^2$ to get the result.

Answer 2

There is an easier proof. By the fundamental theorem of arithmetic, $$a=\prod_{i=1}^{i=n}p_i^{a_i}$$ and $$b=\prod_{i=1}^{i=n}p_i^{b_i}$$ $a_i$ and $b_i$ can be $0$.

Now, $$lcm(a,b)=\prod_{i=1}^{i=n}p_i^{\max(a_i,b_i)}$$ while $$gcd(a,b)=\prod_{i=1}^{i=n}p_i^{\min(a_i,b_i)}$$ Thus, lcm(a,b)*gcd(a,b)=ab.

Since gcd(a,b)=1, lcm(a,b)=ab.