What is the fastest way to figure out what LCD these numbers have in common?

8, 12 and 15. I know that it is 120 but it took me forever to figure it out! I don't want to have to take forever if I have a question like this on my test tonight. Help please =)What is the fastest way to figure out what LCD these numbers have in common?
use prime factorization





8 = 2x2x2


12 = 2x2x3


15 = 3x5





we have 2;3 and 5


lets deal with 2 first. which row the number 2 repeat the most? first row, right?


so you have 2x2x2





lets deal with 3. Which row number 3 repeat the most? since the second row and the third row has number 3 repeat only once.


so you have:


2x2x2x3





lets deal with 5. which row the number 5 repeat the most? the third row. so you have:





2x2x2x3x5 = 120





well, that is the only way to find the LCM. i did it without using a calculatorWhat is the fastest way to figure out what LCD these numbers have in common?
You can always break a number down into a product of a unique set of prime numbers. Just take any factors and break those factors down further until you get all prime numbers:





8 = 2*4 = 2 * 2 * 2


12 = 2*6 = 2 * 2 * 3


15 = 3*5





First, consider the LCD for just 8 and 12. A numer that divides 8 is going to have to at least have ';2*2*2'; in its prime factorization. If this number is to divide 12, then it has to have ';2*2*3'; in its prime factorization too. The smallest way to do this is to simply have ';2*2*2*3'; as the prime factorization, because if our number already has two 2s, all it needs is one 3. Likewise, if this number is going to be divisible by 15 too, then it needs a 3 and 5 for factors. We already have a 3, so just add a 5. Therefore, the LCD is going to be 2*2*2*3*5, or 120.





This is one of those processes that's a little easier to do than to explain, but hopefully you get the idea.
i dont think there is a fast way but heres how:





To find the least common denominator, simply list the multiples of each denominator (multiply by 2, 3, 4, etc.) then look for the smallest number that appears in each list.





Example: Suppose we wanted to add 1/5 + 1/6 + 1/15. We would find the least common denominator as follows...





First we list the multiples of each denominator.





Multiples of 5 are 10, 15, 20, 25, 30, 35, 40,...





Multiples of 6 are 12, 18, 24, 30, 36, 42, 48,...





Multiples of 15 are 30, 45, 60, 75, 90,....





Now, when you look at the list of multiples, you can see that 30 is the smallest number that appears in each list.





Therefore, the least common denominator of 1/5, 1/6 and 1/15 is 30.


This method works pretty good. But, adding fractions with larger numbers in the denominators it can get pretty messy.





So hold that thought for a moment, as we look at another way to find a least common denominator for adding these same fractions.