LCM and GCF
Least Common Multiple
Greatest Common Factor
Definition: A number is prime only if 1 and itself are factors.
The number is prime if it is only divisible by 1 and itself.
List of prime numbers:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, …
Prime Factorization.
Find all the prime numbers that were Multiplied together to produce the original number. In other
words, break down the original number into factors. Continue to break down the factor so that all that
is left is prime number.
Find the prime factorization of 28.
28
2 14
28 = 2•2•7
36
2
153
7
120
LCM: The Least Common Multiple is the smallest multiple that two or more numbers
divide into evenly.
A multiple of a number is the product of the original number with another number.
Some multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, …
Some multiples of 7 are 7, 14, 21, 28, 35, 42, 49, 56, …
A Common Multiple is a number that is divisible by two or more numbers.
Some common multiples of 4 and 7 are 28, 56, 84, and 112.
When looking for the Least Common Multiple, you are looking for the smallest multiple that they both
divide into evenly. The least common multiple of 4 and 7 is 28.
To find least common multiples, you should break all numbers down into their prime factorization.
First consider the COMMON factors they all contain. And then use all the “LEFTOVER” factors.
By multiplying all these factors together you will produce the LCM.
The prime factorization of 4 is 2•2
The prime factorization of 7 is 7
Since there are no common factors, use all the factors (what I call the “leftovers” ),
and multiply them together to produce the LCM, which is 28.
Find the LCM for the following sets of numbers.
24
LCM:
and
42
36
and
LCM:
60
To find least common multiples, you should break all numbers down into their prime factorization.
First consider the COMMON factors they all contain. When there are more than two numbers,
also consider the COMMON factors that some of the numbers contain. And finally, use all the
“LEFTOVER” factors. By multiplying all these factors together you will produce the LCM.
12,
18, and
40
45,
80, and
120
GCF: A Greatest Common Factor is the largest number that is a factor of two or more
numbers.
When looking for the Greatest Common Factor, you are only looking for the COMMON factors
contained in both numbers. If there are no such factors that are common to both, then 1 is the
greatest common factor.
The Greatest Common Factor for 36 and 54 is 18.
The prime factorization for 36 is 2•2•3•3.
The prime factorization for 54 is 2•3•3•3.
They both have in common the factors 2, 3, 3 and their product is 18.
That is why the greatest common factor for 36 and 54 is 18.
24
and
2*2*2*3
42
36
2*3*7
GCF: 2 • 3
6
12,
2*2*3
GCF:
18, and
2*3*3
and
2*2*3*3
60
2*2*3*5
GCF:
40
45,
2*2*2*5
3*3*5
GCF:
80, and
120
2*2*2*2*5 2*2*2*3*5
The similarity between LCM and GCF is that they both use Common factors.
The difference between LCM and GCF this that only the LCM uses all the factors to produce the
Least Common Multiple, even the factors that were not common.