When $\gcd(a,b,c)\cdot \text{lcm}(a,b,c)=\sqrt{abc}$

Question

Recently, I have found this problem:

Given three integer numbers $a,b,c$ such that $1\leq a,b,c\leq 30$ and the following relation holds: $$\gcd(a,b,c)\cdot \text{lcm}(a,b,c)=\sqrt{abc}$$ How many different tuples $(a,b,c)$ are there?

To solve this, I thought to write: $$\text{lcm}(a,b)\cdot c=\gcd(\text{lcm}(a,b),c)\cdot \text{lcm}(\text{lcm}(a,b),c)$$ And: $$\gcd(a,b)\cdot c=\gcd(\gcd(a,b),c)\cdot \text{lcm}(\gcd(a,b),c)$$ So, I have: $$\frac{ab\cdot c^2}{\gcd(\text{lcm}(a,b),c)\cdot \text{lcm}( (\gcd(a,b),c)}=\sqrt{abc}$$ But here I am stuck. Any idea of how to proceed?

Thank you.

Answer 1

(I'm ignoring the triples which contain $0$.)

I would take a different tack and think about the prime factorizations of $a, b,$ and $c$. Let

$$a = p_1^{a_1}\cdots p_k^{a_k}$$

$$b = p_1^{b_1}\cdots p_k^{b_k}$$

$$c = p_1^{c_1}\cdots p_k^{c_k}$$

be the prime factorizations, where some of the exponents might be zero (so we can use the same set of primes for each factorization. If you square both sides of your equation and plug these in you have

$$\prod_{i=1}^{k} p_i^{2\max\{a_i,b_i,c_i\} + 2\min\{a_i,b_i,c_i\} } = \prod_{i=1}^{k} p_i^{a_i+b_i+c_i}. $$

So for each $i$ you must have

$$2\max\{a_i,b_i,c_i\} + 2\min\{a_i,b_i,c_i\} = a_i+b_i+c_i.$$

At this point, WLOG, suppose $a_i \leq b_i \leq c_i.$ Then the last equation is

$$2 c_i + 2a_i = a_i+b_i+c_i.$$

Or

$$ c_i + a_i = b_i.$$

This can only be true if $a_i = 0$ and $c_i = b_i.$

So we have this principle: If a prime divides any of $a, b, c$ then it divides exactly two of them and to the same power. Try $a=5, b=10, c=2$. Yup, it works. Try $17, 17, 1$. Yup. Try $5, 6, 30.$ Yup.

So here's the plan: WLOG, assume $a$ is the smallest member of the triple, then let $a$ count from $1$ to $30.$

If $a=1$, it forces $b=c$ and all choices for $b$ work. So that's 30 solutions.

If $a=2$, exactly one of $b$ and $c$ is exactly divisible by $2$. Say $2\mid b$. Then any prime power that divides $c$ must divide $b$, so we have $b= 2c$, with $c$ odd. That gives us 7 more solutions as $c$ counts the odds from $3$ to $15.$ (We skip $c=1$ because it has to be at least as large as $a$.

If $a=3$, we, similarly, look at the triples $(3, 3c, c)$ where $3\leq c\leq 10$, and $3\nmid c$. We add $(3,12,4), (3, 15, 5), (3,21,7), (3, 24, 8), (3, 30,10).$

If $a=4$, then $4 \mid b$, say and $2\nmid c$ and any prime dividing $c$ must divide $b$, so again we add $(4,20,5), (4,28,7).$ Because $c$ must be odd and greater than $4$ and $b=4c\leq 30.$

If $a=5$, we look at triples $(5, 5c, c).$ Here, $c > 5$ but $5c\leq 30$. So the only triple has $c=6$. Add $(5, 30, 6)$ to the list.

If $a=6$, we have cases. If $6\mid b$ then $\gcd(6,c)=1$ and $c\geq 7$. The only possibilities for $c$ are the primes greater than $6$. Each of these would have to divide $b$ which makes $b$ too large. So no more solutions here.

If $2 \mid b$ and $3\mid c$ then $c$ must be odd and $b$ must not be divisible by $3$. Otherwise, $b$ and $c$ must share the same primes, so there is a number $m$ such that $b = 2m$ and $c=3m$ and $\gcd(6,m)=1$. Since $c\leq 30$, the only possibilites for $m$ are $5$ and $7$. Add $(6,10,15)$ and $(6,14,21)$ to the list.

If $a$ is prime power bigger than $6$, then the triple is $(a, ac, c)$, but $c\geq a$, so $ac \geq 49 >30.$ so no solutions. We have eliminated $a = 7,8,9, 11, 13, 16, 17, 19, 23, 25, 27, 29.$

If $a=2p$ with $p$ and odd prime-power greater than $4$ then we have the same cases as for $a=6$. Either the triple is $(2p, 2pn,n)$ with $n\geq 2p$ which makes $b\geq 4p^2$ but this is too large. So no further solutions. The other case has $(2p, 2n, pn).$ But here, $p$ and $n$ are both at least $5$ and $n$ has to be odd and different from $p$. So $np$ is at least $35$. No solutions here. We have elimnated $10, 14, 18, 22, 26.$

Similarly, if $a=4p$ with $p$ an odd prime-power greater than $4$ there are no solutions. Cross off $20,$ and $28$. If $a=8p$, same thing. Cross off $24$.

If $a= 3p$ with $p=5$ or $7$, then we have cases again. First we might have $(3p, 3pn, n)$ this forces $b$ to be too large. Second, we might have $(3p, 3n, pn)$. This forces $c$ to be too large. Cross off $15$ and $21$.

If $a=12$, then $b=4n$ and $c=3n$ for some integer $n$ less than $30/4$ and relatively prime to $12$. So $n=5$ or $7$ giving us two more solutions. $(12, 15, 20)$ and $(12, 21, 28)$.

All that's left if $a=30$, which forces $b=c=30$ which is not a solution.

Answer 2

Not a 'real' answer, but it was too big for a comment.

I wrote and ran some Mathematica code:

In[1]:=Length[Solve[{GCD[a, b, c]*LCM[a, b, c] == Sqrt[a*b*c], 
   0 <= a <= b <= c <= 30}, {a, b, c}, Integers]]

Running the code gives:

Out[1]=545

Looking for the solutions, we can see:

In[2]:=FullSimplify[
 Solve[{GCD[a, b, c]*LCM[a, b, c] == Sqrt[a*b*c], 
   0 <= a <= b <= c <= 30}, {a, b, c}, Integers]]

Out[2]={{a -> 0, b -> 0, c -> 0}, {a -> 0, b -> 0, c -> 1}, {a -> 0, b -> 0, 
  c -> 2}, {a -> 0, b -> 0, c -> 3}, {a -> 0, b -> 0, 
  c -> 4}, {a -> 0, b -> 0, c -> 5}, {a -> 0, b -> 0, 
  c -> 6}, {a -> 0, b -> 0, c -> 7}, {a -> 0, b -> 0, 
  c -> 8}, {a -> 0, b -> 0, c -> 9}, {a -> 0, b -> 0, 
  c -> 10}, {a -> 0, b -> 0, c -> 11}, {a -> 0, b -> 0, 
  c -> 12}, {a -> 0, b -> 0, c -> 13}, {a -> 0, b -> 0, 
  c -> 14}, {a -> 0, b -> 0, c -> 15}, {a -> 0, b -> 0, 
  c -> 16}, {a -> 0, b -> 0, c -> 17}, {a -> 0, b -> 0, 
  c -> 18}, {a -> 0, b -> 0, c -> 19}, {a -> 0, b -> 0, 
  c -> 20}, {a -> 0, b -> 0, c -> 21}, {a -> 0, b -> 0, 
  c -> 22}, {a -> 0, b -> 0, c -> 23}, {a -> 0, b -> 0, 
  c -> 24}, {a -> 0, b -> 0, c -> 25}, {a -> 0, b -> 0, 
  c -> 26}, {a -> 0, b -> 0, c -> 27}, {a -> 0, b -> 0, 
  c -> 28}, {a -> 0, b -> 0, c -> 29}, {a -> 0, b -> 0, 
  c -> 30}, {a -> 0, b -> 1, c -> 1}, {a -> 0, b -> 1, 
  c -> 2}, {a -> 0, b -> 1, c -> 3}, {a -> 0, b -> 1, 
  c -> 4}, {a -> 0, b -> 1, c -> 5}, {a -> 0, b -> 1, 
  c -> 6}, {a -> 0, b -> 1, c -> 7}, {a -> 0, b -> 1, 
  c -> 8}, {a -> 0, b -> 1, c -> 9}, {a -> 0, b -> 1, 
  c -> 10}, {a -> 0, b -> 1, c -> 11}, {a -> 0, b -> 1, 
  c -> 12}, {a -> 0, b -> 1, c -> 13}, {a -> 0, b -> 1, 
  c -> 14}, {a -> 0, b -> 1, c -> 15}, {a -> 0, b -> 1, 
  c -> 16}, {a -> 0, b -> 1, c -> 17}, {a -> 0, b -> 1, 
  c -> 18}, {a -> 0, b -> 1, c -> 19}, {a -> 0, b -> 1, 
  c -> 20}, {a -> 0, b -> 1, c -> 21}, {a -> 0, b -> 1, 
  c -> 22}, {a -> 0, b -> 1, c -> 23}, {a -> 0, b -> 1, 
  c -> 24}, {a -> 0, b -> 1, c -> 25}, {a -> 0, b -> 1, 
  c -> 26}, {a -> 0, b -> 1, c -> 27}, {a -> 0, b -> 1, 
  c -> 28}, {a -> 0, b -> 1, c -> 29}, {a -> 0, b -> 1, 
  c -> 30}, {a -> 0, b -> 2, c -> 2}, {a -> 0, b -> 2, 
  c -> 3}, {a -> 0, b -> 2, c -> 4}, {a -> 0, b -> 2, 
  c -> 5}, {a -> 0, b -> 2, c -> 6}, {a -> 0, b -> 2, 
  c -> 7}, {a -> 0, b -> 2, c -> 8}, {a -> 0, b -> 2, 
  c -> 9}, {a -> 0, b -> 2, c -> 10}, {a -> 0, b -> 2, 
  c -> 11}, {a -> 0, b -> 2, c -> 12}, {a -> 0, b -> 2, 
  c -> 13}, {a -> 0, b -> 2, c -> 14}, {a -> 0, b -> 2, 
  c -> 15}, {a -> 0, b -> 2, c -> 16}, {a -> 0, b -> 2, 
  c -> 17}, {a -> 0, b -> 2, c -> 18}, {a -> 0, b -> 2, 
  c -> 19}, {a -> 0, b -> 2, c -> 20}, {a -> 0, b -> 2, 
  c -> 21}, {a -> 0, b -> 2, c -> 22}, {a -> 0, b -> 2, 
  c -> 23}, {a -> 0, b -> 2, c -> 24}, {a -> 0, b -> 2, 
  c -> 25}, {a -> 0, b -> 2, c -> 26}, {a -> 0, b -> 2, 
  c -> 27}, {a -> 0, b -> 2, c -> 28}, {a -> 0, b -> 2, 
  c -> 29}, {a -> 0, b -> 2, c -> 30}, {a -> 0, b -> 3, 
  c -> 3}, {a -> 0, b -> 3, c -> 4}, {a -> 0, b -> 3, 
  c -> 5}, {a -> 0, b -> 3, c -> 6}, {a -> 0, b -> 3, 
  c -> 7}, {a -> 0, b -> 3, c -> 8}, {a -> 0, b -> 3, 
  c -> 9}, {a -> 0, b -> 3, c -> 10}, {a -> 0, b -> 3, 
  c -> 11}, {a -> 0, b -> 3, c -> 12}, {a -> 0, b -> 3, 
  c -> 13}, {a -> 0, b -> 3, c -> 14}, {a -> 0, b -> 3, 
  c -> 15}, {a -> 0, b -> 3, c -> 16}, {a -> 0, b -> 3, 
  c -> 17}, {a -> 0, b -> 3, c -> 18}, {a -> 0, b -> 3, 
  c -> 19}, {a -> 0, b -> 3, c -> 20}, {a -> 0, b -> 3, 
  c -> 21}, {a -> 0, b -> 3, c -> 22}, {a -> 0, b -> 3, 
  c -> 23}, {a -> 0, b -> 3, c -> 24}, {a -> 0, b -> 3, 
  c -> 25}, {a -> 0, b -> 3, c -> 26}, {a -> 0, b -> 3, 
  c -> 27}, {a -> 0, b -> 3, c -> 28}, {a -> 0, b -> 3, 
  c -> 29}, {a -> 0, b -> 3, c -> 30}, {a -> 0, b -> 4, 
  c -> 4}, {a -> 0, b -> 4, c -> 5}, {a -> 0, b -> 4, 
  c -> 6}, {a -> 0, b -> 4, c -> 7}, {a -> 0, b -> 4, 
  c -> 8}, {a -> 0, b -> 4, c -> 9}, {a -> 0, b -> 4, 
  c -> 10}, {a -> 0, b -> 4, c -> 11}, {a -> 0, b -> 4, 
  c -> 12}, {a -> 0, b -> 4, c -> 13}, {a -> 0, b -> 4, 
  c -> 14}, {a -> 0, b -> 4, c -> 15}, {a -> 0, b -> 4, 
  c -> 16}, {a -> 0, b -> 4, c -> 17}, {a -> 0, b -> 4, 
  c -> 18}, {a -> 0, b -> 4, c -> 19}, {a -> 0, b -> 4, 
  c -> 20}, {a -> 0, b -> 4, c -> 21}, {a -> 0, b -> 4, 
  c -> 22}, {a -> 0, b -> 4, c -> 23}, {a -> 0, b -> 4, 
  c -> 24}, {a -> 0, b -> 4, c -> 25}, {a -> 0, b -> 4, 
  c -> 26}, {a -> 0, b -> 4, c -> 27}, {a -> 0, b -> 4, 
  c -> 28}, {a -> 0, b -> 4, c -> 29}, {a -> 0, b -> 4, 
  c -> 30}, {a -> 0, b -> 5, c -> 5}, {a -> 0, b -> 5, 
  c -> 6}, {a -> 0, b -> 5, c -> 7}, {a -> 0, b -> 5, 
  c -> 8}, {a -> 0, b -> 5, c -> 9}, {a -> 0, b -> 5, 
  c -> 10}, {a -> 0, b -> 5, c -> 11}, {a -> 0, b -> 5, 
  c -> 12}, {a -> 0, b -> 5, c -> 13}, {a -> 0, b -> 5, 
  c -> 14}, {a -> 0, b -> 5, c -> 15}, {a -> 0, b -> 5, 
  c -> 16}, {a -> 0, b -> 5, c -> 17}, {a -> 0, b -> 5, 
  c -> 18}, {a -> 0, b -> 5, c -> 19}, {a -> 0, b -> 5, 
  c -> 20}, {a -> 0, b -> 5, c -> 21}, {a -> 0, b -> 5, 
  c -> 22}, {a -> 0, b -> 5, c -> 23}, {a -> 0, b -> 5, 
  c -> 24}, {a -> 0, b -> 5, c -> 25}, {a -> 0, b -> 5, 
  c -> 26}, {a -> 0, b -> 5, c -> 27}, {a -> 0, b -> 5, 
  c -> 28}, {a -> 0, b -> 5, c -> 29}, {a -> 0, b -> 5, 
  c -> 30}, {a -> 0, b -> 6, c -> 6}, {a -> 0, b -> 6, 
  c -> 7}, {a -> 0, b -> 6, c -> 8}, {a -> 0, b -> 6, 
  c -> 9}, {a -> 0, b -> 6, c -> 10}, {a -> 0, b -> 6, 
  c -> 11}, {a -> 0, b -> 6, c -> 12}, {a -> 0, b -> 6, 
  c -> 13}, {a -> 0, b -> 6, c -> 14}, {a -> 0, b -> 6, 
  c -> 15}, {a -> 0, b -> 6, c -> 16}, {a -> 0, b -> 6, 
  c -> 17}, {a -> 0, b -> 6, c -> 18}, {a -> 0, b -> 6, 
  c -> 19}, {a -> 0, b -> 6, c -> 20}, {a -> 0, b -> 6, 
  c -> 21}, {a -> 0, b -> 6, c -> 22}, {a -> 0, b -> 6, 
  c -> 23}, {a -> 0, b -> 6, c -> 24}, {a -> 0, b -> 6, 
  c -> 25}, {a -> 0, b -> 6, c -> 26}, {a -> 0, b -> 6, 
  c -> 27}, {a -> 0, b -> 6, c -> 28}, {a -> 0, b -> 6, 
  c -> 29}, {a -> 0, b -> 6, c -> 30}, {a -> 0, b -> 7, 
  c -> 7}, {a -> 0, b -> 7, c -> 8}, {a -> 0, b -> 7, 
  c -> 9}, {a -> 0, b -> 7, c -> 10}, {a -> 0, b -> 7, 
  c -> 11}, {a -> 0, b -> 7, c -> 12}, {a -> 0, b -> 7, 
  c -> 13}, {a -> 0, b -> 7, c -> 14}, {a -> 0, b -> 7, 
  c -> 15}, {a -> 0, b -> 7, c -> 16}, {a -> 0, b -> 7, 
  c -> 17}, {a -> 0, b -> 7, c -> 18}, {a -> 0, b -> 7, 
  c -> 19}, {a -> 0, b -> 7, c -> 20}, {a -> 0, b -> 7, 
  c -> 21}, {a -> 0, b -> 7, c -> 22}, {a -> 0, b -> 7, 
  c -> 23}, {a -> 0, b -> 7, c -> 24}, {a -> 0, b -> 7, 
  c -> 25}, {a -> 0, b -> 7, c -> 26}, {a -> 0, b -> 7, 
  c -> 27}, {a -> 0, b -> 7, c -> 28}, {a -> 0, b -> 7, 
  c -> 29}, {a -> 0, b -> 7, c -> 30}, {a -> 0, b -> 8, 
  c -> 8}, {a -> 0, b -> 8, c -> 9}, {a -> 0, b -> 8, 
  c -> 10}, {a -> 0, b -> 8, c -> 11}, {a -> 0, b -> 8, 
  c -> 12}, {a -> 0, b -> 8, c -> 13}, {a -> 0, b -> 8, 
  c -> 14}, {a -> 0, b -> 8, c -> 15}, {a -> 0, b -> 8, 
  c -> 16}, {a -> 0, b -> 8, c -> 17}, {a -> 0, b -> 8, 
  c -> 18}, {a -> 0, b -> 8, c -> 19}, {a -> 0, b -> 8, 
  c -> 20}, {a -> 0, b -> 8, c -> 21}, {a -> 0, b -> 8, 
  c -> 22}, {a -> 0, b -> 8, c -> 23}, {a -> 0, b -> 8, 
  c -> 24}, {a -> 0, b -> 8, c -> 25}, {a -> 0, b -> 8, 
  c -> 26}, {a -> 0, b -> 8, c -> 27}, {a -> 0, b -> 8, 
  c -> 28}, {a -> 0, b -> 8, c -> 29}, {a -> 0, b -> 8, 
  c -> 30}, {a -> 0, b -> 9, c -> 9}, {a -> 0, b -> 9, 
  c -> 10}, {a -> 0, b -> 9, c -> 11}, {a -> 0, b -> 9, 
  c -> 12}, {a -> 0, b -> 9, c -> 13}, {a -> 0, b -> 9, 
  c -> 14}, {a -> 0, b -> 9, c -> 15}, {a -> 0, b -> 9, 
  c -> 16}, {a -> 0, b -> 9, c -> 17}, {a -> 0, b -> 9, 
  c -> 18}, {a -> 0, b -> 9, c -> 19}, {a -> 0, b -> 9, 
  c -> 20}, {a -> 0, b -> 9, c -> 21}, {a -> 0, b -> 9, 
  c -> 22}, {a -> 0, b -> 9, c -> 23}, {a -> 0, b -> 9, 
  c -> 24}, {a -> 0, b -> 9, c -> 25}, {a -> 0, b -> 9, 
  c -> 26}, {a -> 0, b -> 9, c -> 27}, {a -> 0, b -> 9, 
  c -> 28}, {a -> 0, b -> 9, c -> 29}, {a -> 0, b -> 9, 
  c -> 30}, {a -> 0, b -> 10, c -> 10}, {a -> 0, b -> 10, 
  c -> 11}, {a -> 0, b -> 10, c -> 12}, {a -> 0, b -> 10, 
  c -> 13}, {a -> 0, b -> 10, c -> 14}, {a -> 0, b -> 10, 
  c -> 15}, {a -> 0, b -> 10, c -> 16}, {a -> 0, b -> 10, 
  c -> 17}, {a -> 0, b -> 10, c -> 18}, {a -> 0, b -> 10, 
  c -> 19}, {a -> 0, b -> 10, c -> 20}, {a -> 0, b -> 10, 
  c -> 21}, {a -> 0, b -> 10, c -> 22}, {a -> 0, b -> 10, 
  c -> 23}, {a -> 0, b -> 10, c -> 24}, {a -> 0, b -> 10, 
  c -> 25}, {a -> 0, b -> 10, c -> 26}, {a -> 0, b -> 10, 
  c -> 27}, {a -> 0, b -> 10, c -> 28}, {a -> 0, b -> 10, 
  c -> 29}, {a -> 0, b -> 10, c -> 30}, {a -> 0, b -> 11, 
  c -> 11}, {a -> 0, b -> 11, c -> 12}, {a -> 0, b -> 11, 
  c -> 13}, {a -> 0, b -> 11, c -> 14}, {a -> 0, b -> 11, 
  c -> 15}, {a -> 0, b -> 11, c -> 16}, {a -> 0, b -> 11, 
  c -> 17}, {a -> 0, b -> 11, c -> 18}, {a -> 0, b -> 11, 
  c -> 19}, {a -> 0, b -> 11, c -> 20}, {a -> 0, b -> 11, 
  c -> 21}, {a -> 0, b -> 11, c -> 22}, {a -> 0, b -> 11, 
  c -> 23}, {a -> 0, b -> 11, c -> 24}, {a -> 0, b -> 11, 
  c -> 25}, {a -> 0, b -> 11, c -> 26}, {a -> 0, b -> 11, 
  c -> 27}, {a -> 0, b -> 11, c -> 28}, {a -> 0, b -> 11, 
  c -> 29}, {a -> 0, b -> 11, c -> 30}, {a -> 0, b -> 12, 
  c -> 12}, {a -> 0, b -> 12, c -> 13}, {a -> 0, b -> 12, 
  c -> 14}, {a -> 0, b -> 12, c -> 15}, {a -> 0, b -> 12, 
  c -> 16}, {a -> 0, b -> 12, c -> 17}, {a -> 0, b -> 12, 
  c -> 18}, {a -> 0, b -> 12, c -> 19}, {a -> 0, b -> 12, 
  c -> 20}, {a -> 0, b -> 12, c -> 21}, {a -> 0, b -> 12, 
  c -> 22}, {a -> 0, b -> 12, c -> 23}, {a -> 0, b -> 12, 
  c -> 24}, {a -> 0, b -> 12, c -> 25}, {a -> 0, b -> 12, 
  c -> 26}, {a -> 0, b -> 12, c -> 27}, {a -> 0, b -> 12, 
  c -> 28}, {a -> 0, b -> 12, c -> 29}, {a -> 0, b -> 12, 
  c -> 30}, {a -> 0, b -> 13, c -> 13}, {a -> 0, b -> 13, 
  c -> 14}, {a -> 0, b -> 13, c -> 15}, {a -> 0, b -> 13, 
  c -> 16}, {a -> 0, b -> 13, c -> 17}, {a -> 0, b -> 13, 
  c -> 18}, {a -> 0, b -> 13, c -> 19}, {a -> 0, b -> 13, 
  c -> 20}, {a -> 0, b -> 13, c -> 21}, {a -> 0, b -> 13, 
  c -> 22}, {a -> 0, b -> 13, c -> 23}, {a -> 0, b -> 13, 
  c -> 24}, {a -> 0, b -> 13, c -> 25}, {a -> 0, b -> 13, 
  c -> 26}, {a -> 0, b -> 13, c -> 27}, {a -> 0, b -> 13, 
  c -> 28}, {a -> 0, b -> 13, c -> 29}, {a -> 0, b -> 13, 
  c -> 30}, {a -> 0, b -> 14, c -> 14}, {a -> 0, b -> 14, 
  c -> 15}, {a -> 0, b -> 14, c -> 16}, {a -> 0, b -> 14, 
  c -> 17}, {a -> 0, b -> 14, c -> 18}, {a -> 0, b -> 14, 
  c -> 19}, {a -> 0, b -> 14, c -> 20}, {a -> 0, b -> 14, 
  c -> 21}, {a -> 0, b -> 14, c -> 22}, {a -> 0, b -> 14, 
  c -> 23}, {a -> 0, b -> 14, c -> 24}, {a -> 0, b -> 14, 
  c -> 25}, {a -> 0, b -> 14, c -> 26}, {a -> 0, b -> 14, 
  c -> 27}, {a -> 0, b -> 14, c -> 28}, {a -> 0, b -> 14, 
  c -> 29}, {a -> 0, b -> 14, c -> 30}, {a -> 0, b -> 15, 
  c -> 15}, {a -> 0, b -> 15, c -> 16}, {a -> 0, b -> 15, 
  c -> 17}, {a -> 0, b -> 15, c -> 18}, {a -> 0, b -> 15, 
  c -> 19}, {a -> 0, b -> 15, c -> 20}, {a -> 0, b -> 15, 
  c -> 21}, {a -> 0, b -> 15, c -> 22}, {a -> 0, b -> 15, 
  c -> 23}, {a -> 0, b -> 15, c -> 24}, {a -> 0, b -> 15, 
  c -> 25}, {a -> 0, b -> 15, c -> 26}, {a -> 0, b -> 15, 
  c -> 27}, {a -> 0, b -> 15, c -> 28}, {a -> 0, b -> 15, 
  c -> 29}, {a -> 0, b -> 15, c -> 30}, {a -> 0, b -> 16, 
  c -> 16}, {a -> 0, b -> 16, c -> 17}, {a -> 0, b -> 16, 
  c -> 18}, {a -> 0, b -> 16, c -> 19}, {a -> 0, b -> 16, 
  c -> 20}, {a -> 0, b -> 16, c -> 21}, {a -> 0, b -> 16, 
  c -> 22}, {a -> 0, b -> 16, c -> 23}, {a -> 0, b -> 16, 
  c -> 24}, {a -> 0, b -> 16, c -> 25}, {a -> 0, b -> 16, 
  c -> 26}, {a -> 0, b -> 16, c -> 27}, {a -> 0, b -> 16, 
  c -> 28}, {a -> 0, b -> 16, c -> 29}, {a -> 0, b -> 16, 
  c -> 30}, {a -> 0, b -> 17, c -> 17}, {a -> 0, b -> 17, 
  c -> 18}, {a -> 0, b -> 17, c -> 19}, {a -> 0, b -> 17, 
  c -> 20}, {a -> 0, b -> 17, c -> 21}, {a -> 0, b -> 17, 
  c -> 22}, {a -> 0, b -> 17, c -> 23}, {a -> 0, b -> 17, 
  c -> 24}, {a -> 0, b -> 17, c -> 25}, {a -> 0, b -> 17, 
  c -> 26}, {a -> 0, b -> 17, c -> 27}, {a -> 0, b -> 17, 
  c -> 28}, {a -> 0, b -> 17, c -> 29}, {a -> 0, b -> 17, 
  c -> 30}, {a -> 0, b -> 18, c -> 18}, {a -> 0, b -> 18, 
  c -> 19}, {a -> 0, b -> 18, c -> 20}, {a -> 0, b -> 18, 
  c -> 21}, {a -> 0, b -> 18, c -> 22}, {a -> 0, b -> 18, 
  c -> 23}, {a -> 0, b -> 18, c -> 24}, {a -> 0, b -> 18, 
  c -> 25}, {a -> 0, b -> 18, c -> 26}, {a -> 0, b -> 18, 
  c -> 27}, {a -> 0, b -> 18, c -> 28}, {a -> 0, b -> 18, 
  c -> 29}, {a -> 0, b -> 18, c -> 30}, {a -> 0, b -> 19, 
  c -> 19}, {a -> 0, b -> 19, c -> 20}, {a -> 0, b -> 19, 
  c -> 21}, {a -> 0, b -> 19, c -> 22}, {a -> 0, b -> 19, 
  c -> 23}, {a -> 0, b -> 19, c -> 24}, {a -> 0, b -> 19, 
  c -> 25}, {a -> 0, b -> 19, c -> 26}, {a -> 0, b -> 19, 
  c -> 27}, {a -> 0, b -> 19, c -> 28}, {a -> 0, b -> 19, 
  c -> 29}, {a -> 0, b -> 19, c -> 30}, {a -> 0, b -> 20, 
  c -> 20}, {a -> 0, b -> 20, c -> 21}, {a -> 0, b -> 20, 
  c -> 22}, {a -> 0, b -> 20, c -> 23}, {a -> 0, b -> 20, 
  c -> 24}, {a -> 0, b -> 20, c -> 25}, {a -> 0, b -> 20, 
  c -> 26}, {a -> 0, b -> 20, c -> 27}, {a -> 0, b -> 20, 
  c -> 28}, {a -> 0, b -> 20, c -> 29}, {a -> 0, b -> 20, 
  c -> 30}, {a -> 0, b -> 21, c -> 21}, {a -> 0, b -> 21, 
  c -> 22}, {a -> 0, b -> 21, c -> 23}, {a -> 0, b -> 21, 
  c -> 24}, {a -> 0, b -> 21, c -> 25}, {a -> 0, b -> 21, 
  c -> 26}, {a -> 0, b -> 21, c -> 27}, {a -> 0, b -> 21, 
  c -> 28}, {a -> 0, b -> 21, c -> 29}, {a -> 0, b -> 21, 
  c -> 30}, {a -> 0, b -> 22, c -> 22}, {a -> 0, b -> 22, 
  c -> 23}, {a -> 0, b -> 22, c -> 24}, {a -> 0, b -> 22, 
  c -> 25}, {a -> 0, b -> 22, c -> 26}, {a -> 0, b -> 22, 
  c -> 27}, {a -> 0, b -> 22, c -> 28}, {a -> 0, b -> 22, 
  c -> 29}, {a -> 0, b -> 22, c -> 30}, {a -> 0, b -> 23, 
  c -> 23}, {a -> 0, b -> 23, c -> 24}, {a -> 0, b -> 23, 
  c -> 25}, {a -> 0, b -> 23, c -> 26}, {a -> 0, b -> 23, 
  c -> 27}, {a -> 0, b -> 23, c -> 28}, {a -> 0, b -> 23, 
  c -> 29}, {a -> 0, b -> 23, c -> 30}, {a -> 0, b -> 24, 
  c -> 24}, {a -> 0, b -> 24, c -> 25}, {a -> 0, b -> 24, 
  c -> 26}, {a -> 0, b -> 24, c -> 27}, {a -> 0, b -> 24, 
  c -> 28}, {a -> 0, b -> 24, c -> 29}, {a -> 0, b -> 24, 
  c -> 30}, {a -> 0, b -> 25, c -> 25}, {a -> 0, b -> 25, 
  c -> 26}, {a -> 0, b -> 25, c -> 27}, {a -> 0, b -> 25, 
  c -> 28}, {a -> 0, b -> 25, c -> 29}, {a -> 0, b -> 25, 
  c -> 30}, {a -> 0, b -> 26, c -> 26}, {a -> 0, b -> 26, 
  c -> 27}, {a -> 0, b -> 26, c -> 28}, {a -> 0, b -> 26, 
  c -> 29}, {a -> 0, b -> 26, c -> 30}, {a -> 0, b -> 27, 
  c -> 27}, {a -> 0, b -> 27, c -> 28}, {a -> 0, b -> 27, 
  c -> 29}, {a -> 0, b -> 27, c -> 30}, {a -> 0, b -> 28, 
  c -> 28}, {a -> 0, b -> 28, c -> 29}, {a -> 0, b -> 28, 
  c -> 30}, {a -> 0, b -> 29, c -> 29}, {a -> 0, b -> 29, 
  c -> 30}, {a -> 0, b -> 30, c -> 30}, {a -> 1, b -> 1, 
  c -> 1}, {a -> 1, b -> 2, c -> 2}, {a -> 1, b -> 3, 
  c -> 3}, {a -> 1, b -> 4, c -> 4}, {a -> 1, b -> 5, 
  c -> 5}, {a -> 1, b -> 6, c -> 6}, {a -> 1, b -> 7, 
  c -> 7}, {a -> 1, b -> 8, c -> 8}, {a -> 1, b -> 9, 
  c -> 9}, {a -> 1, b -> 10, c -> 10}, {a -> 1, b -> 11, 
  c -> 11}, {a -> 1, b -> 12, c -> 12}, {a -> 1, b -> 13, 
  c -> 13}, {a -> 1, b -> 14, c -> 14}, {a -> 1, b -> 15, 
  c -> 15}, {a -> 1, b -> 16, c -> 16}, {a -> 1, b -> 17, 
  c -> 17}, {a -> 1, b -> 18, c -> 18}, {a -> 1, b -> 19, 
  c -> 19}, {a -> 1, b -> 20, c -> 20}, {a -> 1, b -> 21, 
  c -> 21}, {a -> 1, b -> 22, c -> 22}, {a -> 1, b -> 23, 
  c -> 23}, {a -> 1, b -> 24, c -> 24}, {a -> 1, b -> 25, 
  c -> 25}, {a -> 1, b -> 26, c -> 26}, {a -> 1, b -> 27, 
  c -> 27}, {a -> 1, b -> 28, c -> 28}, {a -> 1, b -> 29, 
  c -> 29}, {a -> 1, b -> 30, c -> 30}, {a -> 2, b -> 3, 
  c -> 6}, {a -> 2, b -> 5, c -> 10}, {a -> 2, b -> 7, 
  c -> 14}, {a -> 2, b -> 9, c -> 18}, {a -> 2, b -> 11, 
  c -> 22}, {a -> 2, b -> 13, c -> 26}, {a -> 2, b -> 15, 
  c -> 30}, {a -> 3, b -> 4, c -> 12}, {a -> 3, b -> 5, 
  c -> 15}, {a -> 3, b -> 7, c -> 21}, {a -> 3, b -> 8, 
  c -> 24}, {a -> 3, b -> 10, c -> 30}, {a -> 4, b -> 5, 
  c -> 20}, {a -> 4, b -> 7, c -> 28}, {a -> 5, b -> 6, 
  c -> 30}, {a -> 6, b -> 10, c -> 15}, {a -> 6, b -> 14, 
  c -> 21}, {a -> 12, b -> 15, c -> 20}, {a -> 12, b -> 21, c -> 28}}

So, we can see that when we have $(\text{a},\text{b},\text{c})$ where $0\le\text{a}\le\text{b}\le\text{c}\le30$ there are $545$ solutions to that problem.

Answer 3

Suppose $(a,b,c)$ is a solution with $abc\ne 0$ and $a\le b\le c$, and suppose $p$ is a prime with $p^r||a$, $p^s||b$, $p^t||c$ ($||$ means that the quotient is not divisible by $p$). After renaming $r$, $s$, $t$ we may assume $r\le s\le t$. Then the power of $p$ in $\sqrt{abc}$ is $\frac{r+s+t}{2}$ while the power of $p$ in $\gcd(a,b,c)\cdot\mathrm{lcm}(a,b,c)$ is $r+t$, so that $r+t=s$. But $r\le s\le t$ then implies that $r=0$ and $s=t$. Therefore $\gcd(a,b,c)=1$. Further, this implies that if $p$ divides any of $a$, $b$, and $c$, then it divides exactly two of them, and to the same power.

Now, given $a$ and $b$ satisfying that condition (that is, that if a prime divides both $a$ and $b$, it divides them to the same power), it is easy to construct the only $c$ that works: take the product of the prime power factors unique to $a$ and $b$. Thus for example if $a = 8\cdot 27$ and $b = 27\cdot 25$, then we take $c=8\cdot 25$.

Answer 4

This is not a solution but maybe a way you can use to continue your analysis. We assume a,b,c>0.

We have $$\gcd(a,b,c)\cdot \text{lcm}(a,b,c)=\sqrt{abc}$$ and $a$,$b$,$c$ are products of prime powers so this must also hold for the prime powers. We have $$\gcd(p^u,p^v,p^w)\text{lcm}(p^u,p^v,p^w)=\sqrt{p^u p^v p^w}$$ or $$\min(u,v,w)+\max(u,v,w)=\frac{u+v+w}2$$ Without loss of generality we assume $u\le v \le w$ and we get $$u=0, v=w$$

So for a prime $p$ and a power $e$ such that $p^e<30$ we have the triples $(a,b,c)=$ $$(1,p^e,p^e),(p^e,1,p^e),(p^e,p^e,1)$$ that satisfy the conditions. If we have two such triples $(a_1,b_1,c_1)$ and $(a_2,b_2,c_2)$ such that no prime divides $a_1a_2$,$b_1 b_2$ and $c_1c_2$ and that $a_1 a_2\le 30$,$b_1 b_2\le 30$,$c_1c_2\le 30$, then $(a_1 a_2,b_1 b_2,c_1c_2)$, is a solution, too. So let's construct some solutions:

$2^2\le 30$, so $(2^2,1,2^2)$ is a solution

$5^1\le 30$, so $(5^1,5^1,1)$ is a solution

and also $(2^2 5^1,5^1,2^2 )=(20,5,4)$

In a similar manner we find out that

$(2^1 5^1,3^1 5^1,2^1 3^1 )=(10,15,6)$