Counting number of friendly triplets

Question

A triplet (a,b,c) is called friendly if product of any two is equal to third number . Find number of such ordered triplet. a,b,c are real numbers

Answer 1

Assuming that you are looking for triples $(a,b,c)$ of real numbers such that $ab=c$ and $bc=a$ and $ca=b$, then multiply all three equalities to get

$$ab\cdot bc \cdot ca = abc$$ $$(abc)^2=abc$$

The only two real solutions of $x^2=x$ are $x=0$ and $x=1$. So, either $abc=0$ or $abc=1$.

If $abc=0$, then one of $a$,$b$ or $c$ is zero. Without loss of generality, assume $a=0$. Then, $b=ca=0$ and $c=ab=0$.

If $abc=1$, substitute $bc=a$ to get $a^2=1$. Similarly, $b^2=1$ and $c^2=1$. This means $a=\pm 1$, $b=\pm 1$ and $c=\pm 1$. This gives 8 possible triples $(a,b,c)$ which can be checked by hand.