Least Common Multiple

Khushboo · 2007-12-31 00:35:51

I am currently developing activities on Least Common Multiple. My objective is to make this topic interactive so i am trying to explore this topic as much as possible. Request you to give feedback and add some ideas to the existing pool of activities i plan to create.

Least common multiple is basically union of the two numbers. We can thus use factor tree to achieve the union. Say for instance the union of 6 & 9 is 18 which is also the least common multiple.

One method can be creating a two dimensional array dynamically. As you create the array for two numbers and try to equate them, you will achieve the least common multiple.

the least common multiple can be achieved by skip counting. Say for numbers 8 and 5 , we need to do skip count by 8 &5 simultaneously. As we do counting we will get the lowest common multiple which is 40

Even bar diagram can be used effectively to explain lcm.

Request you to add and provide feedback.

Regards

Khushboo

Jai Ganesh · 2007-12-31 20:09:33

This is how we were taught at school to find the LCM.
Start with 2, and then 3, 5, 7, 11, ...so on (only prime numbers).
Divide all the numbers by the marked number. If it is not divisible, write down the same number.
If 2 is a common factor of 2 or more numbers, try 2 again. Similarly, for other prime numbers.
At the end, multiply all the numbers in the grid.
This is done only when there are no more common prime factors.

For example, finding the LCM of 180 and 216.
Divide the two numbers by 2. You get 90 and 108.
Divide again by 2, you get 45 and 54.
Divide by 3, you get 15 and 18.
Divide again by 3, you get 5 and 6.
There are no more prime factors in common.
Hence multiply
2x2x3x3x5x6:- you get 1080.
This is the LCM.
180 six times is 1080.
216 five times is 1080.
Hence, 1080 is the LCM.

Jai Ganesh · 2007-12-31 20:16:10

Finding the Greatest Common Divisor or the Greatest Common Factor or the Highest Common Factor is almost the same. Only difference being, the numbers at the bottom of the grid aren't multiplied to get the result.
Hence, in the above example, that is, the Greatest Common Divisor of 180 and 216 would be 2x2x3x3 = 36.

mathsyperson · 2008-01-01 02:12:28

Ganesh's method is my favourite for when the numbers are small enough that splitting them into prime factors is a possibility.

When it's not, Euclid's algorithm is an excellent tool. It involves writing the bigger of your two numbers in terms of a multiple of the smaller one, plus a remainder. The highest common factor (HCF) of the smaller number and the remainder is the same as the HCF of the original two, and so you can use that fact iteratively to bring the two numbers down to a managable level.

In ganesh's example, we would have 216 = 1x180 + 36.
Therefore hcf(180, 216) = hcf(36, 180).
180 is a multiple of 36, and so their hcf is 36.

After finding the hcf in this way, the lcd follows quite easily due to this fact:
hcf(a,b) x lcd(a,b) = a x b, for any two a and b.

That might be a bit more complicated than you were looking for, but I like it because it always works, no matter what order of magnitude your numbers are.
The bolded fact is very useful even if you don't use Euclid's algorithm beforehand.

Khushboo · 2008-01-01 02:53:39

Hi

Let me first wish you guy's a very happy new year. I hope year 2008 be the memorable year for all of us and we scale to new heights.

Thanx a lot for sharing the lcm act. Will get back soon with more lcm act and also let me see how i can develop them.

Regards

Khushboo

Math Is Fun Forum

#1 2007-12-31 00:35:51

Least Common Multiple

#2 2007-12-31 20:09:33

Re: Least Common Multiple

#3 2007-12-31 20:16:10

Re: Least Common Multiple

#4 2008-01-01 02:12:28

Re: Least Common Multiple

#5 2008-01-01 02:53:39

Re: Least Common Multiple

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