Some Notes on the Least Common Multiple
In class, we saw that the least common multiple (or LCM for short) of the denominators of two
fractions served as a common denominator for the fractions, so that we could add the fractions. We also
saw a method for finding the LCM of two numbers: for each of the numbers, give a list of its multiples
– the first number common to both lists is the LCM.
11
, we find the LCM of 8 and 18:
For example, to find the sum 83 + 18
Multiples of 8: 8, 16, 24, 36, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120, 128, 136, 144, 152, ...
Multiples of 18: 18, 36, 54, 72, 90, 108, 126, 144, 162, 180, 198, ...
From the list, we see that the LCM is 72. We then convert each fraction to the LCM and add:
3
8
+
11
18
= 38 ( 99 ) +
11 4
( )
18 4
=
27
72
+
44
72
=
71
72
One advantage to this method of finding the LCM is that it is fairly direct – we keep writing down
multiples until we find an overlap on the lists. One disadvantage to this method is that we often may
have to write down many multiples before we find a common multiple. Another disadvantage is that we
will want to find LCMs of polynomials later on, and this method is hard to translate into a method for
polynomials (how could we even list all of the multiples of a polynomial?).
Another method for finding the LCM involves factoring. Recall that a prime number is a natural
number that is not divisible by any smaller natural number, except for 1. For example, 2, 3, and 5
are prime numbers, while 4 and 6 are not (4 is divisible by 2; 6 is divisible by 2 and 3). We think
of prime numbers as basic multiplication building blocks in the world of natural numbers. The prime
factorization of a natural number writes the number as a product of prime numbers.
Here are some examples of prime factorizations:
4 = 2 ∗ 2 = 22
5 = 5 (5 itself is prime)
6=2∗3
8 = 2 ∗ 2 ∗ 2 = 23
18 = 2 ∗ 3 ∗ 3 = 2 ∗ 32
21 = 3 ∗ 7
Note that 8 = 4 ∗ 2 is not a prime factorization of 8, since 4 is not a prime.
To find the prime factorization of a natural number n, find a prime p by which n is divisible. Divide
n by p to get another smaller number m, so that n = p ∗ m. Repeat this process with m to factor
another prime out, and continue. For example, let’s find the prime factorization of 60:
A prime that divides 60 is 5. Dividing out, we get 60 = 5 ∗ 12. Now we need to break down 12.
A prime that divides 12 is 2. Dividing gives 12 = 2 ∗ 6, so 60 = 5 ∗ 2 ∗ 6. Now we break down 6.
A prime that divides 6 is 2. We get 6 = 2 ∗ 3, so 60 = 5 ∗ 2 ∗ 2 ∗ 3. Since 2, 3, and 5 are all primes,
this is a prime factorization. We can tidy this up a bit by grouping together 2 ∗ 2 as 22 and putting the
primes in increasing order: 60 = 22 ∗ 3 ∗ 5.
Prime factorizations provide us another method for finding the LCM of two numbers. To find the
LCM of two numbers, find all of the primes occuring in the prime factorization of either number. For
each such prime choose the larger power that it has between the two factorizations (remember that if
we write a number without an exponent, then its power is 1). Then, the LCM is the product of these
numbers.
Here are some examples:
To find the LCM of 4 and 6, write their prime factorizations: 4 = 22 , 6 = 2 ∗ 3. The primes occuring are
2 and 3. The largest power of 2 occuring is the second power, 22 (in the factorization of 4). The largest
power of 3 occuring is the first power, 3 (in the factorization of 6). The LCM is the product of these:
22 ∗ 3 = 12.
To find the LCM of 4 and 8, write their prime factorizations: 4 = 22 , 8 = 23 . The only prime occuring
is 2. The largest power of 2 occuring is 23 (in the factorization of 8). The LCM is then 23 = 8.
To find the LCM of 6 and 7, write their prime factorizations: 6 = 2 ∗ 3, 7 = 7 (7 itself is prime).
The primes occuring are 2, 3, and 7. The largest power of 2 occuring is the first power, 2 (in the
factorization of 6). The largest power of 3 occuring is the first power, 3 (in the factorization of 6). The
largest power of 7 occuring is the first power, 7 (in the factorization of 7). The LCM is the product of
these: 2 ∗ 3 ∗ 7 = 42.
To find the LCM of 8 and 18, write their prime factorizations: 8 = 23 , 18 = 2 ∗ 32 . The primes occuring
are 2 and 3. The largest power of 2 occuring is the third power, 23 (in the factorization of 8). The
largest power of 3 occuring is the second power, 32 (in the factorization of 18). The LCM is the product
of these: 23 ∗ 32 = 72.
To find the LCM of 18 and 40, write their prime factorizations: 18 = 2 ∗ 32 , 40 = 23 ∗ 5. The primes
occuring are 2, 3, and 5. The largest power of 2 occuring is the third power, 23 (in the factorization of
40). The largest power of 3 occuring is the second power, 32 (in the factorization of 18). The largest
power of 5 occuring is the first power, 5 (in the factorization of 40). The LCM is the product of these:
23 ∗ 32 ∗ 5 = 360.
In this example, to find the LCM by the old method would require us to write out at least twenty
multiples of 18 and nine multiples of 40!
Try this method yourself:
1)
2)
3)
4)
5)
6)
7)
Find the LCM of
Find the LCM of
Find the LCM of
Find the LCM of
Find the LCM of
6
4
+ 25
?
What is 15
11
7
What is 15 + 54 ?
4 and 5.
6 and 18.
6 and 21.
15 and 25.
15 and 54.
(Use #4 to find a common denominator)