An ordered triple of real numbers

(a,b,c) is called friendly, if each
number is equal to the product of
the other 2. How many friendly
triples are there?
Details and assumptions
The numbers need not be equal to
each other.
The numbers need not to be pairwise distinct (which
means that no two of them
are the same). I got 31 but it isn't right ...

To find the number of friendly triples, we can start by listing out all possible values for (a, b, c) and checking if they satisfy the given conditions.

We know that each number in the triple is equal to the product of the other two. Let's use a systematic approach to list out the possible values:

Case 1: a = 0
If we let a be 0, then we have b*c = 0. In this case, there are infinite possible values for (b, c) since either b or c can be 0, and the other can take any real value.

Case 2: a ≠ 0
If a is not equal to 0, then we have b*c = a. In this case, we can choose any value for a, and then find the corresponding values of (b, c).

Let's summarize the possibilities:

1. If a = 0, then there are infinite possibilities for (b, c).
2. If a ≠ 0, then we can choose any real value for a, and calculate the corresponding values for (b,c).

From case 1 alone, we already have infinite possibilities. However, if we consider case 2, we can see that for each value of a, there is one unique solution for (b, c) since they are determined by the value of a.

Therefore, the number of friendly triples is infinite.

Hence, the answer to the question is that there are infinitely many friendly triples satisfying the given conditions.