What is the difference between greatest common factor and a multiple

The greatest common factor GCF is the largest number that is a factor of two or more numbers, and the least common multiple LCM is the smallest number that is a multiple of two or more numbers. Before we can add fractions, we have to make sure the denominators are the same by creating an equivalent fraction:.

In this example, the least common multiple of 3 and 6 must be determined. In the context of adding or subtracting fractions, the least common multiple is referred to as the least common denominator.

In general, you need to determine a number larger than or equal to two or more numbers to find their least common multiple. It is important to note that there is more than one way to determine the least common multiple. One way is to simply list all the multiples of the values in question and select the smallest shared value, as seen here:. This illustrates that the least common multiple of 8, 4, and 6 is 24 because it is the smallest number that 8, 4, and 6 can all divide into evenly.

Another common method involves the prime factorization of each value. Remember, a prime number is only divisible by 1 and itself. Once the prime factors are determined, list the shared factors once, and then multiply them by the other remaining prime factors. The result is the least common multiple:. The least common multiple can also be found by common or repeated division.

This method is sometimes considered faster and more efficient than listing multiples and finding prime factors. Here is an example of finding the least common multiple of 3, 6, and 9 using this method:. Divide the numbers by the factors of any of the three numbers.

Repeat this process until all of the numbers are reduced to 1. Then, multiply all of the factors together to get the least common multiple. We will be identifying a value smaller than or equal to the numbers being considered.

Prime factorization can also be used to determine the greatest common factor. However, rather than multiplying all the prime factors like we did for the least common multiple, we will multiply only the prime factors that the numbers share. The resulting product is the greatest common factor. Index Newest Popular Best. New Player Log In. Newest Questions Post a Question Search All Questions Please cite any factual claims with citation links or references from authoritative sources.

Editors continuously recheck submissions and claims. Archived Questions Goto Qn. LauraMcC Answer has 4 votes. LauraMcC 14 year member replies Answer has 4 votes. Currently voted the best answer. The lowest common multiple LCM is the lowest number that is a multiple of two numbers.

However, to avoid overduplication, the LCM does not need three copies, because neither nor contains three copies. This over-duplication issue with factors often causes confusion, so let's spend a little extra time on this.

Consider two smaller numbers, 4 and 8 , and their LCM. The LCM needs only have three copies of 2 , in order to be divisible by both 4 and 8. That is, the LCM is 8. You do not need to take the three copies of 2 from the 8 , and then throw in two extra copies from the 4. This would give you While 32 is a common multiple, because 4 and 8 both divide evenly into 32 , 32 is not the LEAST smallest common multiple, because you'd have over-duplicated the 2 s when you threw in the extra copies from the 4.

Let me stress again: let the nice neat listing keep track of things for you, especially when the numbers get big. Returning to the exercise:. So, my LCM of and must contain both copies of the factor 2.

By the same reasoning, the LCM must contain both copies of 3 , both copies of 5 , and both copies of 7 :. By using this "factor" method of listing the prime factors neatly in a table, you can always easily find the LCM and GCF. Completely factor the numbers you are given, list the factors neatly with only one factor for each column you can have 2 s columns, 3 s columns, etc, but a 3 would never go in a 2 s column , and then carry the needed factors down to the bottom row.

For the GCF, you carry down only those factors that all the listings share; for the LCM, you carry down all the factors, regardless of how many or few values contained that factor in their listings. For instance, to find the LCM of 4 and 6 , you'd list their multiples, starting with the smallest and working your way up, until you found the first duplicate.

This first duplicate multiple would be your LCM:. But this "listing" method would be awful for large numbers like what we just did above. To find: the unknown integer. Ask your question. Get your answer.

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