Finding at which point multiplied numbers coalesce

[ignorant2]

Hello!

This is not really a homework question, more of a question of what I should do for "homework", but I hope it is acceptable.

My question is regarding a method of finding at which point (if at all) two different numbers, when multiplied, reach the same number. For example, how could I find out that the first common whole number that numbers 1.5 and 2 both share is 6 - 1.5 multiplied by 4 and 2 multiplied by 3.

It should be obvious from my question and (lack of) terminology that I know almost nothing about mathematics. I am looking to learn more and will start a course soon, but in the meantime it would help a lot if someone could tell me what this type of method is called and what would be the prerequisite knowledge for understanding it.

Many thanks!

 

[haruspex]

Start with your fraction 1.5 but with some unknown other number, x. Suppose they both divide a whole number, n. So n = some whole multiple, y, of 1.5. Is there anything you can deduce about y?

 

[RUber]

Let's assume that you are working with Rational numbers (those that can be represented as a fraction).
Your first number may have the form (a/b) and the second may have the form (c/d).
If your goal is to get to whole numbers, you should know that both numbers will only form whole numbers at multiples of b or d, respectively.
If one of your numbers is irrational, (no fractional representation), then you are out of luck.
I would look for solutions to the equation:
$bx(\frac ab )=dy(\frac cd )$.
You might be able to find a general rule for x and y. If you want to find the smallest whole number, you will then need to minimize your x and y.
In your example,
1.5 = 3/2
2 = 2/1
$2x(\frac 32 )=1y(\frac 21 )$
$3x=2y$
$\frac xy= \frac 23$
So x = 2 and y = 3 is a natural choice...your multipliers are 2x2 and 1x3, or 4 and 3 as you pointed out.