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Finding at which point multiplied numbers coalesce

  1. Dec 5, 2014 #1
    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!
     
  2. jcsd
  3. Dec 5, 2014 #2

    haruspex

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    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?
     
  4. Dec 5, 2014 #3

    RUber

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    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.
     
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