Least Common Multiple (LCM)

Let you have some cardinal numbers: "a", "b", "c", then you always can find their product: a × b × c = m.

Now you have another number - "m", which you can divide into "a", into "b", into "c".

Look,

m ⁄ a = a × b × c ⁄ a = b × c
m ⁄ b = a × b × c ⁄ b = a × c
m ⁄ c = a × b × c ⁄ c = a × b

So, this number "m" is called as "common multiple" for numbers "a", "b", "c".

It's clear, you always can find not the only "common multiple" for any numbers. If you choose the least among them, you'll get Least Common Multiple (LCM) for your numbers.

Example:

Find LCM for 270, 300, 315.
270 = 2 × 3 × 3 × 3 × 5 = 2 × 3³ × 5
300 = 2 × 2 × 3 × 5 × 5 = 2² × 3 × 5²
315 = 3 × 3 × 5 × 7 = 3² × 5 × 7
LCM = 2² × 3³ × 5² × 7 = 189000

This example shows you the way for getting "Least Common Multiple" (LCM) for any numbers.

You see, it's enough to find all prime multipliers for every number, then you have to make a product of them, putting in it common multipliers only with the greatest exponents.

If you get "Least Common Multiple" (LCM) and "Greatest Common Divisor" (GCD) easily, then you can always do addition and subtraction for any fractions without mistakes.

Hope, you see now, mathematics is the game which is worth the candle, isn't it?