# > Integers

### Integers

• 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.