
The integers are the numbers 1, 2, 3, and so on, together with their negatives, 1, 2, 3,… , and 0. Thus the set of integers is {…., 3, 2, 1, 0, 1, 2, 3, ….}.

Positive integers are greater than 0, negative integers are less than 0 and 0 is neither positive nor negative. When integers are added, subtracted, or multiplied the result is always an integer.

The product of two positive integers is a positive integer.

The product of two negative integers is a positive integer.

The product of a positive integer and a negative integer is a negative integer.

When integers are multiplied, each of the multiplied integers is called a factor or divisor of the resulting product. For e.g. (2)(3)(10)= 60, so 2, 3, and 10 are factors of 60. The negatives of these integers are also factors of 60, for e.g. (2)(30)= 60. There are no other factors of 60. We say that 60 is a multiple of each of its factors and is divisible by each of its divisors.

The least common multiple of two nonzero integers a and b is the least positive integer that is a multiple of both a and b. For e.g. the least common multiple of 30 and 75 is 150.

The greatest common divisor (or greatest common factor) of two nonzero integers a and b is the greatest positive integer that is a divisor of both a and b. For e.g. the greatest divisor of 30 and 75 is 15.

When an integer a is divided by an integer b, where b is a divisor of a, the result is always a divisor of a. For e.g. when 60 is divided by 6 (one of its divisors), the result is 10, which is another divisor of 60.

If b is not a divisor of a, then the result can be viewed in three different ways. The result can be viewed as a fraction or as a decimal, both of which are discussed later, or the result can be viewed as a quotient with a remainder, where both are integers. For eg. the integers 19 and 7. When 19 is divided by 7, the result is greater than 2, since (2)(7) < 19, but less than 3, since 19 < (3)(7). Because 19 is 5 more than (2)(7), we say that the result of 19 divided by 7 is the quotient 2 with remainder 5, or simply "2 remainder 5." In general, when a positive integer a is divided by a positive integer b, the greatest multiple of b that is less than or equal to a can be expressed as the product qb, where q is the quotient. Then the remainder, r = a – qb.

If an integer is divisible by 2, it is called an even integer; otherwise it is an odd integer.

The sum of two even integers is an even integer.

The sum of two odd integers is an even integer.

The sum of an even integer and an odd integer is an odd integer.

The product of two even integers is an even integer.

The product of two odd in integers is an odd integer.

The product of an even integer and an odd integer is an even integer.

A prime number is an integer greater than 1 that has only two positive divisors: 1 and itself. The first ten prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29. For eg. 14 is not a prime number, since it has four positive divisors: 1, 2, 7 and 14. The integer 1 is not a prime number and the integer 2 is the only prime number that is even.

Every integer greater than 1 either is a prime number or can be uniquely expressed as a product of a factors that are prime numbers, or prime divisors. Such an expression is called a prime factorization. For e.g. 1,155 = (3)(5)(7)(11).

An integer greater than 1 that is not a prime number is called a composite number. The first ten composite numbers are 4, 6, 8, 9, 10, 12, 14, 15, 16, and 18.