**Basics of Integers**

**Click here for definitions of terms**

An *integer* is a whole number, which can be positive, negative, or zero. For example, 5, 17, -8, -365, and 0 are integers. Fractions like 3/4 and -1/2 are not whole numbers, and are therefore not integers.

**Properties of Integers**

*Even integers* are divisible by 2. Examples: 0, 2, 4, 6, 8. *Odd integers* are not evenly divisible by 2. Examples: 1, 3, 5, 7.

*Consecutive integers* are a sequence of integers, each element of which is 1 greater than the previous element. Examples:

0, 1, 2, 3, 4, …

-3, -2, -1, 0, 1, 2, …

*Consecutive even integers* are a sequence of even integers, each element of which is 2 greater than the previous element. Examples:

8, 10, 12, 14, 16, …

-4, -2, 0, 2, 4, 6, …

*Consecutive odd integers* are a sequence of odd integers, each element of which is 2 greater than the previous element. Examples:

7, 9, 11, 13, 15, …

-5, -3, -1, 1, 3, …

A *prime number* is an integer greater than 1 which can only be evenly divided by itself or by 1. Examples: 2, 3, 5, 17, 43. A *factor* is a component of a multiplication product. For example, 2 and 3 are factors of 6, since 6 = 2 x 3. It’s often valuable to break an integer into its prime factors (see *prime factorization*, below).

*Divisiblity*: This states a relationship between two integers. a is divisible by b, if for some integer n, a = bn. In this case, b and n are *divisors* of a, and a is a *multiple* of b and of n.

**Operations With Integers**

The following are common operations used to solve integer problems.

*Prime Factorization:* The representation of a positive integer as a product of powers of *primes*. For example, the prime factorization of 180 is

\(\begin{align} 2^2 \times 3^2 \times 5\end{align}\)

since the only prime numbers that divide 180 are 2 (twice), 3 (twice), and 5. Every integer greater than 1 can be expressed uniquely as a product of powers of primes.

*Greatest Common Factor (Divisor) (gcf or gcd):* The largest number which evenly divides two or more numbers. The greatest common factor (or the greatest common divisor) of 20 and 24 (written gcf(20, 24)), for example, is 4. Even though 2 evenly divides both numbers, it’s smaller than 4, so 4 is the *greatest* common factor.

*Least Common Multiple (lcm):* The smallest number which is a multiple of two or more numbers. For example, the least common multiple of 6 and 10 (written lcm(6, 10)) is 30, since even though 60 is a multiple of both numbers, 30 is also and is smaller than 60. The least common multiple is useful in finding a common denominator for adding two fractions. For example, in adding 1/4 and 1/6 we seek a common denominator. The least common multiple of 4 and 6 is 12, and this is a common denominator that makes it possible to add 1/4 and 1/6:

1/4 + 1/6 =

3/12 + 2/12 = 5/12

Both the greatest common factor and the least common multiple of two numbers can be determined by taking the *prime factorizations* of the numbers. For example, suppose you want the lcm and gcf of 60 and 42. Then the prime factorizations of these two numbers are:

60 = 2²·3·5

42 = 2·3·7

We obtain the lcm by taking all these prime factors, including the highest power of any prime factor, and omitting duplicates. So

lcm(60, 42) = 2²·3·5·7 = 420

The gcf is obtained by finding the common factors of the two numbers. In our example the only common prime factors of 60 and 42 are 2 and 3:

gcf(60, 42) = 2·3 = 6

*Quotients and Remainders: *If an integer n is not evenly divisible by another integer k, then we can find integers q and r, such that n = qk + r, where q is maximized, and r < k. The integer q is called the *quotient*, and r is the *remainder.* Example: Let n be 25 and k be 7. Then q is 3 and r is 4, because 25 = 3·7 + 4.

*Factorial:* This is the operation of multiplying a positive integer by every positive integer less than or equal to it. For example, 4 factorial (written 4!) is

\(\begin{align} 4 \times 3 \times 2 \times 1 = 24\end{align}\)

Factorials are useful in solving problems of permutations and combinations.