How many ways can you find a LCM?

Question

having problems understanding how to figure out the LCM of real numbers.
Have heard multiple ways of solving, but they are all blending together.
Whats the easiest way you know how to find the LCM

Answer 1

The easiest way for me to solve the LCM is to use Lola May which is you take the two numbers you are using. For example

14 and 20 you draw an L and put the numbers inside the L then pick a number that goes into both 14 and 20 so on the outside of the L you would write 2. Under the L you would write 7 under 14 because 2x7=14 and under the 20 you would write 10

Then you multiply 10x2x7 which= 140

Use the diagram below for help.

Answer 2

The simplest way I know is called the ladder method.

You place a 2 in front, then list all of the numbers after it
Now divide all that will divide by two.
Try two on the second row as well if you get any even answers, and even on the next if still even numbers
You Place a 3 in front, and bring down all answers from division, as well as undivided numbers. Repeat with 3 in front if any still will divide by 3
You place a 5 in front and bring down all answers as above, as well as undivided numbers. Repeat if still divisible by 5
Try 7 in front, and do the same, repeating if needed
Then try 11 in front. Rarely you will need to try 13 , 17 or 19 as well.

When you get a row of all 1's you are done.

Your LCM is the product of all numbers you placed first down the left side.

This sounds hard, but it works faster than it describes.

EXAMPLE : Find the LCM of 2,3,6,8,12,and 15


2 | 2 3 6 8 12 15
2 | 1 3 3 4 6 15
2 | 1 3 3 2 3 15
3 | 1 3 3 1 3 15
5 | 1 1 1 1 1 5
1 | 1 1 1 1 1 1
LCM = 2 x 2 x 2 x 3 x 5 = 120

Answer 3

I can think of two ways.

The first is efficient for fairly small numbers, or numbers that have fairly large common factors. It goes like this:

Method 1:
-------------
1. Let "A" be the larger number and let "B" be the smaller number.
2. If "B" divides "A" evenly, you're done.
3. Otherwise, multiply "A" by 2. If "B" divides evenly into the result, then you're done.
4. Otherwise, multiply "A" by 3. If "B" divides evenly into the result, then you're done.
5. Otherwise, multiply "A" by 4. ...(etc.)...

There's a finite number of steps, because ultimately, if all else fails, you'll multiply A by B, and that is always a common multiple.

The above technique is not efficient for large numbers with small common factors. For example, if A=1000001 and B=1000000, it would take you a million multiplication steps before you discovered that the LCM is 1000001000000.

In that case:

Method 2
------------

LCM = A x B / GCF

where "GCF" is the greatest common factor of A and B. Fortunately, there is a very efficient algorithm for finding the GCF:

1. Let M = A (the larger number); and let N = B (the smaller number).
2. Let R = M mod N (i.e. the remainder when M is divided by N). If R is zero, then the GCF is N, and you're done.
3. Otherwise, let M = N, and let N = R, and go to Step 2.

This algorithm finishes in very few steps (very few "loops" of Steps 2 and 3), even for very large A and B. Example: Say A=124 and B=100.

M = 124, N=100
R = M mod N = 24. Not zero, so:
M=100, N=24
R = M mod N = 4. Not zero, so:
M=24, N=4
R = M mod N = 0. Zero! So the GCF is 4.

And finally, that would make the LCM = 124•100/4 = 3100

Answer 4

The most basic (and foolproof) way of doing this is:
1. Find the prime factors of A
2. Find the prime factors of B
3. Identify any prime factors of B that are not prime factors of A.
4. Multiply A by the factors identified in step 3 to produce a number that has all of the prime factors that occur in either A or B. In other words, this is the LCM of A and B.

Important point to understand in steps 1-3:
When you determine the prime factors of a number, you have to keep track of HOW MANY TIMES each prime factor occurs. For example, 12 has three prime factors: 2, 2, and 3.
The prime number 2 occurs twice, and 3 occurs once.
So if A is 12 and B is 40, you have:
Step 1: Prime factors of A: 2, 2, 3
Step 2: Prime factors of B: 2, 2, 2, 5
Step 3: Prime factors of B that don't occur in A: the number 5 and the third number 2 (i.e., A had only two 2's, but B had three 2's).
Step 4: Multiply A by 2 and 5: 12 x 2 x 5 = 120

Applying this to the numbers used by another responder:
A = 100 = 2 x 2 x 5 x 5
B = 124 = 2 x 2 x 31
31 is the only prime factor of B that is not a prime factor of A.
LCM = 100 x 31 = 3100.

Answer 5

In a lot of cases you can use simpler methods, but the method I describe here is general. It works entirely mechanically in all cases.

First find the greatest common divisor (GCD). Say your numbers are a and b, and the greatest common divisor is c. Then the LCM is:

a*b/c

Of course to do this you need to find the GCD. For finding the GCD of a and b, use the following algorithm:

1) Divide the larger number by the smaller to get a remainder. That is if a>b, then use long division to find integers q,r so that

a = q.b+r

2) If r is not zero, then the GCD of r and b is the same as the GCD of a and b. Thus replace a by b and b by r, and go back to step (1. If r is zero, then b is the GCD.

This method of finding the GCD is called the Euclidean algorithm.

Answer 6

easiest way to find LCM is basically multiplying the two. Alot of the times, this will give you the LCM. Once you have done this, you can check it is the LCM by deviding your answer by 2, and seeing if it still works as an LCM. Works quite well.

Answer 7

Lowest common multiple
Divide the main number by numbers less than 9 to derive an answer such as 1,1