# LCM and GCD equation

Let $a$, $b$, $c$ be three positive integers such that $$\mathrm{lcm}(a,b)\cdot\mathrm{lcm}(b,c)\cdot\mathrm{lcm}(c,a)=a\cdot b\cdot c\cdot \gcd(a,b,c).$$

Given that none of $a$, $b$, $c$ is an integer multiple of any other of $a$, $b$, $c$, find the minimum possible value of $a+b+c$.

So, I was able to find that $\gcd(a,b)\cdot\gcd(b,c)\cdot\gcd(a,c)\cdot\gcd(a,b,c)=abc$ but I wasn't quite sure how to continue on. Some help would be great. Thanks!

## 1 Answer

Let's suppose we have

$$\gcd(a,b)\cdot\gcd(b,c)\cdot\gcd(c,a)\cdot \gcd(a,b,c) = abc$$

and look at a prime $p$ dividing at least one of $a,b,c$.

Suppose $p$ divides at most two of the three, say $p\nmid c$, and $a = p^\alpha\cdot a'$, $b = p^\beta\cdot b'$ with $p\nmid a'b'$. Then on the left hand side, $p$ does only occur in $\gcd(a,b)$, with exponent $\min\{\alpha,\beta\}$. But on the right hand side, it occurs with exponent $\alpha+\beta = \min\{\alpha,\beta\} + \max\{\alpha,\beta\}$, and since the exponents must be equal, it follows that $\max\{\alpha,\beta\} = 0$, contradicting the assumption that $p$ divides at least one of $a,b,c$.

So every prime dividing at least one of $a,b,c$ must divide all three. Let the exponents of $p$ be $\alpha \leqslant \beta \leqslant \gamma$. Then on the left hand side, $p$ occurs with the exponent

$$\alpha + \beta + \alpha + \alpha = 3\alpha + \beta,$$

and on the right it occurs with the exponent $\alpha + \beta + \gamma$. It follows that $\gamma = 2\alpha$.

Furthermore, the condition that none of $a,b,c$ shall be an integer multiple of any other implies that for each pair $(x,y)$ of two of the numbers, there is at least one prime $p_{xy}$ that occurs with a larger exponent in the prime factorisation of $x$ than it occurs in the prime factorisation of $y$.

Any prime that is not one of the $p_{xy}$ can be removed from all of $a,b,c$, and leads to a solution with smaller numbers, and smaller $a+b+c$, so the primes involved in the minimal solution are precisely the

$$\{p_{ab},p_{ac},p_{ba},p_{bc},p_{ca},p_{cb}\}.$$

That set must contain at least two primes, and it contains at most six. It is clear that for the minimal solution, the set must contain the $k$ smallest primes, $2 \leqslant k \leqslant 6$.

Finding the minimal solution is then a small amount of work even brute-forcing.