Basic Arithmetic


The numbers in the set {. . . , –4, –3, –2, –1, 0, 1, 2, 3, 4, . . .} are called integers. Naturally, the integers that are greater than 0 are called positive integers, and those that are less than 0 are called negative integers.


Although 0 is neither positive nor negative, it is an integer.

Consecutive integers are two or more integers written in sequence in which each integer is 1 more than the preceding one. For example:

3, 4

15, 16, 17, 18,

–3, –2, –1, 0, 1, 2

n, n + 1, n + 2, n + 3, . . .

The sum, difference, and product of two integers is always an integer. The quotient of two integers may or may not be an integer. The quotient 86 10 can be expressed as or or 8.6. You can also say that the quotient is 8 and the remainder is 6. How you express this depends upon your point of view. For example, if you want to divide 86 dollars equally among 10 people, you can give each one $8.60 (8.6 dollars). However, if you divide 86 books into 10 piles, each pile will have 8 books, and 6 books will be left over (the remainder).

Whenever you have to answer a question involving remainders, think of piles of books. For example, suppose 3 is the remainder when an integer n is divided by 7. Then n could be 17 (2 piles of 7 with 3 left over), n could be 703 (100 piles of 7 with 3 left over), or n could be 3 more than anymultiple of 7.


Never assume that number means integer. If you are told that 1 < x < 4, of course x could be 2 or 3, but it could also be 2.5, 3.33333, π, , or any of infinitely many other numbers.

If m and n are integers, the following four terms are synonymous:

m is a divisor of nm is a factor of n
n is divisible by mn is a multiple of m

They all mean that when n is divided by m, there is no remainder (or, more precisely, the remainder is 0). For example:

4 is a divisor of 12 4 is a factor of 12
12 is divisible by 4 12 is a multiple of 4

Key Fact A5

If m and n are positive integers and if r is the remainder when n is divided by m, then n is r more than a multiple of m. That is, n = mq + r where q is an integer and 0 r < m.

Key Fact A6

Every integer has a finite set of factors (or divisors) and an infinite set of multiples.

The factors of 12:

–12, –6, –4, –3, –2, –1, 1, 2, 3, 4, 6, 12

The multiples of 12:

. . . , –48, –36, –24, –12, 0, 12, 24, 36, 48, . . .

The only positive divisor of 1 is 1. All other positive integers have at least two positive divisors—1 and itself—and possibly many more. For example, 12 is divisible by 1, 2, 3, 4, 6, and 12, whereas 7 is divisible by only 1 and 7. Positive integers, such as 7, which have exactly two positive divisors, are called prime numbers or primes. The first few primes are

2, 3, 5, 7, 11, 13, 17, 19, 23

Memorize this list—it will come in handy.


1 is not a prime.

Key Fact A7

Every integer greater than 1 that is not a prime can be written as a product of primes.

EXAMPLE 1: To find the prime factorization of 180, write 180 as a product of any two factors; then write any of those factors that are not prime as a product of two factors. Continue this process until all the factors listed are prime.

This factorization is usually written with the primes in increasing order:

The least common multiple (LCM) of two or more integers is the smallest positive integer that is a multiple of each of them. For example, the LCM of 8 and 12 is 24. The greatest common factor (GCF) or greatest common divisor (GCD) of two or more integers is the largest integer that is a factor of each of them. For example, the GCF of 8 and 12 is 4. Note that the product of the GCF and LCM of 8 and 12 is 4 24 = 96, which is also the product of 8 and 12.

Key Fact A8

The product of the GCF and LCM of any two positive integers is equal to the product of the two integers.

Key Fact A9

For any integers m and n:

If either m or n is even, then mn is even.

If both m and n are odd, then mn is odd.

If m and n are both even or both odd, then m + n and mn are even.

If either m or n is even and the other is odd, then m + n and mn are odd.

Don”t Get Confused

The terms odd and even apply only to integers