Find all tuples of positive integers $(a, b, c)$ such that, $$\text{lcm}(a, b, c) = \frac{ab + bc + ca}4$$
The hint is "If $p\, | \,(b + c)$, then $p\, |\, bc$. What does this give?"
I've read the solution here.
But, I'm quite struggling on how to get $\text{lcm}(a,b,c)=bc$ and $a |bc$
Given, $$\text{lcm}(a, b, c) = \frac{ab + bc + ca}4$$
WLOG, Let us consider, $a\ge b\ge c$ i.e. $a$ is largest number among $a,b,c$.
According to LCM properties, $$ a \mid \text{lcm}(a,b,c) {\implies a \mid 4\times\text{lcm}(a,b,c)\\ \implies a \mid (ab + bc + ca)\\ \implies a \mid bc \quad (\text{As, $ab$ and $ca$ is already divisible by $a$})\\ \implies a\mid [\text{lcm} (b,c) \times \text{gcd} (b,c)]\\(\text{product of lcm, gcd of two numbers = the product of the numbers.})\\ \implies a\mid \text{lcm} (b,c) \quad (\text{gcd$(b,c)\le c\le a$ . Hence, lcm must be divisible by $a$})\\ \implies \text{lcm} (a,b,c)=bc}$$