What are Integers

An integer is any number in the set {. . . –3, –2, –1, 0, 1, 2, 3, . . .}. Integers are positive and negative whole numbers.

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. 

Even and Odd Numbers

Even numbers are integers divisible by 2: 2, 4, 6, 8, 10, and so on. Odd numbers are those integers that aren’t divisible by 2: 1, 3, 5, 7, 9, 11, and so on. What’s important to remember for the is what happens to even or odd numbers when you add, subtract, or multiply them by one another.

Rules regarding even and odd numbers for addition and subtraction

  1. When you add or subtract two even integers, your result is an even integer.
  2. When you add or subtract two odd integers, your result is also even.
  3. If you add or subtract an even integer and an odd integer, your result is an odd integer.

Multiplying even and odd integers

  1. When you multiply an even number by an even number, you get an even number.
  2. When you multiply an odd number by an even number, you also get an even number.
  3. The only time you get an odd number is when you multiply an odd number by another odd number.

Division rules are a little more complex because the quotients aren’t always integers, sometimes they are fractions.

  1. When you divide an even integer by an odd integer, you get an even integer or a fraction.
  2. An odd integer divided by another odd integer results in an odd integer or a fraction.
  3. An even integer divided by another even integer can result in either an odd or even quotient, so that’s not very helpful.
  4. When you divide an odd integer by an even one, you always get a fraction; because fractions aren’t integers, the quotient for this scenario is neither odd nor even.

Remembering these rules can be a big time saver when it comes to eliminating answer choices. For example, if you have a multiplication problem involving large even numbers, you know you can eliminate any odd-number answer choices without even doing the calculations.

Positive and Negative Numbers

Positive and negative numbers have their own set of rules regarding operations. 

For multiplying and dividing

  1. When you multiply or divide two positive numbers, the result is positive.
  2. When you multiply or divide two negative numbers, the result is also positive.
  3. Multiplying or dividing a negative number by a positive number gives you a negative result (as does multiplying or dividing a positive number by a negative number).

For adding and subtracting

  1. When you add two positive numbers, your result is a positive number.
  2. When you add two negative numbers, the resulting sum is negative.
  3. When you add a positive number to a negative number, the result is positive when the number with the largest absolute value is positive and negative when the number with the largest absolute value is negative.
  4. If you subtract a negative number from another number, you end up adding the positive version of the negative number to the other number. For example, x – (–3) is the same thing as x + 3.

Mathematical Operations of Integers

If at least one factor of a product of integers is even, then the product is even; otherwise the product is odd. If two integers are both even or both odd, then their sum and their difference are even. Otherwise, their sum and their difference are odd.

Prime Numbers

Every number has 1 and itself as a factor. A number is a prime if it has only these two factors. Thus, prime number is a positive integer that has exactly two different positive divisors, 1 and itself.

For example, 2, 3, 5, 7, 11, and 13 are prime numbers, but 15 is not, since 15 has four different positive divisors, 1, 3, 5, and 15. The number 1 is not a prime number since it has only one positive divisor. Every integer greater than 1 either is prime or can be uniquely expressed as a product of prime factors. For example, 14 = (2)(7), 81 = (3)(3)(3)(3), and 484 = (2)(2)(11)(11).

Consecutive Integers

The numbers –2, –1, 0, 1, 2, 3, 4, 5 are consecutive integers. Consecutive integers can be represented by n, n + 1, n + 2, n + 3, . . . , where n is an integer. The numbers 0, 2, 4, 6, 8 are consecutive even integers, and 1, 3, 5, 7, 9 are consecutive odd integers. Consecutive even integers can be represented by 2n, 2n + 2, 2n + 4, . . . , and consecutive odd integers can be represented by 2n + 1, 2n + 3, 2n + 5, . . . , where n is an integer.

Properties of the integer 0

  • The integer 0 is neither positive nor negative.
  • If n is any number, then n + 0 = n and n × 0 = 0
  • Division by 0 is not defined

Properties of the integer 1

  • If n is any number, then 1 × n = n
  • For any number n ≠ 0, n × (1/n) = 1
  • Multiplying or dividing an expression by 1, in any form, does not change the value of that expression
  • The number 1 is not a prime number since it has only one positive divisor