


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.
The numbers 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. The
LCM of two numbers is, like the GCF, useful when manipulating fractions:
For example, what is the least common multiple of 4 and
6? We must first find their prime factorizations.
Their LCM is the smallest prime factorization that contains
every prime number in each of the two original prime factorizations.
For the numbers 4 and 6, this is 2 2 3
= 12. It is the smallest prime factorization that includes 2 2 3.
Thus, 12 is the LCM of 4 and 6.
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:
Therefore, their LCM is 2 7 19
= 266.
For some quick 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.
