DIVISIBILITY
The things in a set are called elements or members. The union of Set A and Set B, sometimes
expressed as A B, is the set of elements that are in either or both of Set A and Set B. If Set A =
{1, 2} and Set B = {3, 4}, then A B = {1, 2, 3, 4}. The intersection of Set A and Set B, sometimes
expressed as A B, is the set of elements common to both Set A and Set B. If Set A = {1, 2, 3} and
Set B = {3, 4, 5}, then A B = {3}.
8 . Factor/Multiple
The factors of integer n are the positive integers that divide into n with no remainder. The
multiples of n are the integers that n divides into with no remainder. For example, 6 is a factor
of 12, and 24 is a multiple of 12. 12 is both a factor and a multiple of itself, since 12 × 1 = 12 and
12 ÷ 12 = 1.
9 . Prime Factorization
To find the prime factorization of an integer, continue factoring until all the factors are
prime. For example, to factor 36: 36 = 4 × 9 = 2 × 2 × 3 × 3.
10 . Relative Primes
Relative primes are integers that have no common factor other than 1. To determine whether
two integers are relative primes, break them both down to their prime factorizations. For
example, 35 = 5 × 7, and 54 = 2 × 3 × 3 × 3. They have no prime factors in common, so 35 and 54
are relative primes.
11 . Common Multiple
A common multiple of two or more integers is a number that is a multiple of all of these
integers. You can always get a common multiple of two integers by multiplying them, but,
unless the two numbers are relative primes, the product will not be the least common
multiple. For example, to find a common multiple for 12 and 15, you could just multiply: 12 ×
15 = 180.
To find the least common multiple (LCM), test the multiples of the larger integer until you
find one that’s also a multiple of the smaller. To find the LCM of 12 and 15, begin by taking
the multiples of 15: 15 is not divisible by 12; 30 is not; nor is 45. But the next multiple of 15, 60,
is divisible by 12, so it’s the LCM.
