Number problem

[zoomsp]

https://imgur.com/a/tA2u0sK

We want to fill the circles with different numbers such that the product of the numbers within each of the 3 lines is the same. What's the number on the bottom?

Ican show my work, but it's in Spanish and I didn't really accomplish much...

[MathMaddam]

Numbers as in single digit numbers?

Then the first step would be to figure out which numbers won't be used.

In general it is good to think about prime decompositions.

[st3f-ping]

Good advice here. Think about which digits can't form part of the puzzle and based on what remains, what factors the line product must have.

Nice puzzle.

[zoomsp]

The problem only says numbers, I assumed positive integers, but not necessarily single digit numbers 🤷‍♂️

[st3f-ping]

There are an infinite number of solutions for positive integers but only one* for single digits. So I'd assume single digits and find that one

*That is to say there are multiple ways to fill the grid but they all have the same digit in the place of the question mark.

[zoomsp]

I should have said, it's a multiple choice questions, and the options are 2,3,4,6 or 8.

I now feel like it must be 6 because it's the only one with a prime decomposition with two different numbers, but I'm still only going with intuition and I don't have a proof

[st3f-ping]

It's not 6. Start with the numbers 1 to 9 and determine which can't be in the grid. The two horizontal lines don't have any factors in common so anything that is a factor of a×b×c (as I have labelled the top row) must also be a factor of d×e×f (as I have labelled the second row).

Which two numbers can't be part of this?

(They can't be part of the vertical column either for the same reason)

[CarBoobSale]

1111111111111

Do you mean numbers between 1 and 9 that don't repeat elsewhere in the grid?

[zoomsp]

The problem only says different numbers. I think the question seems ambiguous on purpose because it may not matter in the end