1. Properties of Integers An integer is any number in the set {. . . –3, –2, –1, 0, 1, 2, 3, . . .}. If x and y are integers and x ≠ 0, then x is a divisor (factor) of y provided that y = xn for some integer n. In this case, y is also said to be divisible by x or to be a multiple of x. For example, 7 is a divisor or factor of 28 since 28 = (7)(4), but 8 is not a divisor of 28 since there is no integer n such that 28 = 8n.
If x and y are positive integers, there exist unique integers q and r, called the quotient and remainder, respectively, such that y = xq + r and 0 ≤ r < x. For example, when 28 is divided by 8, the quotient is 3 and the remainder is 4 since 28 = (8)(3) + 4. Note that y is divisible by x if and only if the remainder r is 0; for example, 32 has a remainder of 0 when divided by 8 because 32 is divisible by 8. Also, note that when a smaller integer is divided by a larger integer, the quotient is 0 and the remainder is the smaller integer. For example, 5 divided by 7 has the quotient 0 and the remainder 5 since 5 = (7)(0) + 5.