Greatest Common Divisors and Least Common Multiples. First of all, let’s what is a common divisor (or common factor). Well, it is just a
divisor that two or more numbers have in common.
Example:
Find common divisors od 24 and 36.
1st step: list the divisors of each:
24: 1, 2, 3, 4, 6, 8, 12, 24
36: 1, 2, 3, 4, 6, 9, 12, 18, 36
2nd step. Now, circle the ones that they have in common:
24: 1, 2, 3, 4, 6, 8, 12, 24
36: 1, 2, 3, 4, 6, 9, 12, 18, 36
So, the common factors of 24 and 36 are 1,2,3,4,6,12.
Now, what about Greatest Common Divisor (GCD)?
It is very easy to find GCD for relatively small numbers.
Take two numbers from the example above. We just do steps 1 and 2 and then
choose the greatest out of the list of common divisors for these two numbers.
Thus, the GCD for 24 and 36 is 12.
Common Multiples are just the multiples that two or or more numbers have
together.
Example: find some of the common multiples of 2 and 3.
1st step: list multiples of each
2: 2, 4, 6, 8, 10, 12, 14,….
3: 3, 6, 9, 12, 15, …..
2nd step: circle the multiples that they have in common
2: 2, 4, 6, 8, 10, 12, 14,….
3: 3, 6, 9, 12, 15,….
So, the common multiples of 2 and 3 are 6, 12,…
• Now, let me ask you: Can you list all common multiples of 2 and 3?
Now, to find the Least Common Multiple (LCM) you just choose the smallest
one out of the list of common multiples. In our example that would be 6!
But there exist the other way to find the LCM.
But before starting to use method number 2, we need to learn about
prime divisors.
To get to the concept of prime factors we first need to learn the difference
between prime and composite numbers.
Can you find all divisors of 6? 12?
6: 1, 2, 3, 6
12: 1, 2, 3, 4, 6, 12
What about all the divisors of 13? 19?
13: 1,13
19: 1,19
They have exactly 2 factors each: 1 and the number itself!!!!
This kind of numbers we call prime numbers.
Numbers, that have exactly 2 divisors 1 and number itself are called prime
numbers.
A composite number has more than two divisors.
Our next step will be prime factorization.
Ok, we all know that
𝟏𝟐 = 𝟒×𝟑
But can we write it as a product of only prime factors?
Indeed,
𝟏𝟐 = 𝟐×𝟐×𝟑
So, when we write a number as a product of only prime numbers it is called a
prime factorization.
Let’s see how we can use it to find the LCM of three numbers (to make it more
interesting).
Method 2
Example: Find the least common multiple of 5, 6 and 15.
1st step: To use this method factor each of the numbers into primes.
Prime factorization of 6 is 2 x 3
Prime factorization of 15 is 3 x 5
2nd step: Then for each different prime number in all of the factorizations, do the
following...
• Count the number of times each prime number appears in each of the
factorizations.
Notice that the different primes are 2, 3 and 5
The count of primes in 6 is one 2 and one 3
The count of primes in 15 is one 3 and one 5
• For each prime number, take the largest of these counts. So we have...
The largest count of 2s is one
The largest count of 3s is one
The largest count of 5s is one
3rd step: write down that prime number as many times as you counted for it in
step 2.
Here they are...
2, 3, 5
4th step: The least common multiple is the product of all the prime numbers
written down.
𝟐× 𝟑 ×𝟓 = 𝟑𝟎 Therefore, the least common multiple of 6 and 15 is 30.
.
Indeed,
𝟐× 𝟑 ×𝟓 = 𝟑𝟎.
This gives us another variation of method 2.
1. We repeat step 1 of method 2.
2. Circle all prime factors that our two numbers have in common
Prime factorization of 6 is 2 x 3
Prime factorization of 15 is 3 x 5
3. This common part we use once and then add all other factors from the
factorization of the firs number and all factors from the factorization of the
second number
𝟐× 𝟑 ×𝟓 = 𝟑𝟎.
𝟐× 𝟑 ×𝟓 = 𝟑𝟎.