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, ......}.

The positive integers are greater than 0, the 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. A division may or may not yield an integer. The many elementary number facts for these operations, such as \(7 + 8 =15\), \(22 - 33= -11\), \(7 - (-8)= 15\), and \(7 * 8= 56\).

Here are some general facts regarding multiplication of integers.

• 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 example, \((2)(3)(10)= 60\), so 2, 3, and 10 are factors of 60. The integers 4, 15, 5, and 12 are also factors of 60, since \((4)(15)= 60\) and \((5)(12)= 60\). The positive factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. The negatives of these integers are also factors of 60, since, for example, \((-2)(-30)= 60\). There are no other factors of 60. We say that 60 is a multiple of each of its factors and that 60 is divisible by each of its divisors. Here are some more examples of factors and multiples.

• The positive factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, and 100.
• 25 is a multiple of only six integers: 1, 5, 25, and their negatives.
• The list of positive multiples of 25 has no end: 25, 50, 75, 100, 125, 150, etc.; likewise, every nonzero integer has infinitely many multiples.
• 1 is a factor of every integer; 1 is not a multiple of any integer except 1 and 1.
• 0 is a multiple of every integer; 0 is not a factor of any integer except 0.

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 example, the least common multiple of 30 and 75 is 150. This is because the positive multiples of 30 are 30, 60, 90, 120, 150, 180, 210, 240, 270, 300, etc., and the positive multiples of 75 are 75, 150, 225, 300, 375, 450, etc. Thus, the common positive multiples of 30 and 75 are 150, 300, 450, etc., and the least of these 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 example, the greatest common divisor of 30 and 75 is 15. This is because the positive divisors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30, and the positive divisors of 75 are 1, 3, 5, 15, 25, and 75. Thus, the common positive divisors of 30 and 75 are 1, 3, 5, and 15, and the greatest of these 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 example, 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. Each view is useful, depending on the context. Fractions and decimals are useful when the result must be viewed as a single number, while quotients with remainders are useful for describing the result in terms of integers only.

Regarding quotients with remainders, consider two positive integers a and b for which b is not a divisor of a; for example, 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 \(b\), you first find the greatest multiple of \(b\) that is less than or equal to \(a\). That multiple of \(b\) can be expressed as the product \(qb\), where \(q\) is the quotient. Then the remainder is equal to a - that multiple of \(b\), or where \(r= a -qb\), where \(r\) is the remainder. The remainder is always greater than or equal to \(0\) and less than \(b\).