Just know they all mean the same thing — find the greatest positive integer that divides evenly into two or more numbers. Sometimes, they will be used interchangeably. But in all honesty, they require the same math process, so many teachers and students use these two phrases as synonyms. But, regardless of what the technique is called, the process for finding the greatest common factor and the least common multiple is very straightforward.
The numbers also share one copy of 3 , one copy of 5 , and one copy of 7. On the other hand, the Least Common Multiple, the LCM, is the smallest "least" number that both and will divide into.
That is, it is the smallest number that contains both and as factors, the smallest number that is a multiple of both these values; it is the multiple common to the two values. Therefore, it will be the smallest number that contains every factor in these two numbers. Looking back at the listing, I see that has one copy of the factor of 2 ; has two copies. 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. Would you want to try listing multiples for and ? Example: The common factors of 15, 30 and Factors of 15 are 1, 3, 5, and 15 Factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30 Factors of are 1, 3, 5, 7, 15, 21, 35 and The factors that are common to all three numbers are 1, 3, 5 and 15 In other words, the common factors of 15, 30 and are 1, 3, 5 and The "Greatest Common Factor" is the largest of the common factors of two or more numbers.
Example: How can we simplify 12 30? The Greatest Common Factor of 12 and 30 is 6. We can: find all factors of both numbers use the All Factors Calculator , then find the ones that are common to both, and then choose the greatest.
Or we can find the prime factors and combine the common ones together: Two Numbers Thinking
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