Multiples
A multiple is an integer that can divide evenly into 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.
If John dates Chloe just because she’s popular and wealthy, then
the relationship is a multiple of popularity and wealth, and John
is still a total jerk. Here’s a better example: What are some multiples
of 4? The numbers 12, 20, and 96 are all multiples of 4. How do
we know? Because
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.
Also similar to the GCF, you can’t find the LCM without using prime
factorization. For example, what’s the least common multiple of
4 and 6? Begin by prime factorizing:
The LCM—get this, it’s tricky—is equal to the multiplication
of each factor by the maximum number of times it appears in either
number. Since 2 appears twice in the prime factorization of 4, it
will appear twice (2
2) in the LCM.
Since 3 appears once, it will appear once. So the LCM of 4 and 6
is 2
2
3 = 12.
One more example will help. What is the LCM of 14 and
38? Prime factorize:
Since 2 appears a maximum of once in either number, it
will appear once in the LCM. Same goes for 7 and 19, making the
LCM 2
7
19 = 266.
