


Multiples
A multiple is an integer that can be evenly divided by
another integer. If c ⁄d is
an integer, then c is a multiple of d.
45, 27, and 18, for example, are all multiples of 9. Alternatively,
you could define a multiple as an integer with at least one factor.
All that really matters is that you understand the concept of multiples,
and this is best done with a simple example.
What are some multiples of 4?
 12, 20, and 96 are all multiples of 4.
How do we know these numbers are multiples of 4?
Also, note that any integer, n, is a
multiple of 1 and n, because 1 n = n.
Least Common Multiple
The least common multiple (LCM) of two integers is the
smallest multiple that the two numbers have in common. Like the
GCF, the least common multiple of two numbers is useful when manipulating
fractions.
To find the LCM of two integers, you must first find the
integers’ prime factorizations. The least common multiple is the
smallest prime factorization that contains every prime number in
each of the two original prime factorizations. If the same prime
factor appears in the prime factorizations of both integers, multiply
the factor by the greatest number of times it appears in the factorization
of either number.
For example, what is the least common multiple of 4 and
6? We must first find their prime factorizations.
In this example, 2 appears as a prime factor of both integers.
It appears twice in the prime factorization in which it is more
prevalent, so to find the LCM we will use two 2s. We will also use
the 3 from the prime factorization of 6. The LCM of 4 and 6 is therefore
Let’s try a harder example. What is the LCM of 14 and
38? Again, we start by finding the prime factorizations of both
numbers:
In this example, 2 appears in both prime factorizations,
but not more than once in each, so we only need to use one 2. Therefore,
the LCM of 7 and 38 is
For practice, find the LCM of the following pairs of integers:
 12 and 32
 15 and 26
 34 and 40
 3 and 17
 18 and 16
Compare your answers to the solutions:
 12 = 2^{3} 3. 32 = 2^{5}. The LCM is 2^{5} 3 = 96.
 15 = 3 5. 26 = 2 13. The LCM is 2 3 5 13 = 390.
 34 = 2 17. 40 = 2^{3} 5. The LCM is 2^{3} 5 17 = 680.
 3 = 1 3. 17 = 1 17. The LCM is 3 17 = 51.
 18 = 2 3^{2}. 16 = 2^{4}. The LCM is 2^{4} 3^{2} = 144.
