Pairs of Coprime Integers

Because the problem asks for the product of three different numbers, that means that one number will have to be shared by both. Because we're looking for coprime numbers, the only common factor that they can have is 1. Note that 2 and 4 must be in the same product, because otherwise the two numbers will share a common factor of 2. We see that one of the numbers must be $1 \times 2 \times 4$ , which means that the other must be $1 \times 3 \times 5$ . We see that this is the only pair that satisfies the condition, therefore the answer is 1.

Note that the only odd product is $1 \cdot 3 \cdot 5 = 15$ . It is clear that any pair of even integers is not coprime. So, what is left to us is to check which of the other $9$ products are not divisible by $3$ or $5$ . The only product with that condition is $1\cdot 2\cdot 4 = 8$ , and thus, the answer is $\boxed {1}$ .

As 1 of the numbers is going to repeat in the pair of three, it can't be any number other than one or else they won't be relatively prime. Now, if one of the pairs has got 2 in itself, the other one can't have 2 nor can have 4. So the first pair will have to have 4 in itself. So, the only pair formed will be - 1×2×4 and 1×3×5 will be its corresponding pair.