The properties of integers include the properties of Closure, Associative, Commutative, Identity, Inverse, and Distributive for the operations of addition and multiplication. These properties are not true for the operations of subtraction and division.
The set of integers is closed over the operations of addition and multiplication. When two integers are combined using addition or multiplication, the result is always another integer.
The Associative property of addition and multiplication over the system of integers states that the order in which we combine three integer values does not change the result. For example:
 
The Commutative property of addition and multiplication over the system of integers states that the order in which we combine two integer values does not change the result. For example:
The Identity property of addition identifies the numbers that do not change the value of an integer number under the operations of addition and multiplication.

The additive identity is the integer 0. For example, when you add 0 to any integer, the result is the original integer.
The multiplicative identity is the integer 1. Any integer multiplied by 1 will result in the original integer number.
The understanding of the Inverse property is important for integer operations. The Inverse property of addition represents the idea that when combining two integers the result is the additive identity, 0. Therefore, a + b = 0. If this is true then b must be the additive inverse of a, or -a. The Inverse property states that adding an integer and its inverse will result in the additive identity, or 0.
The Distributive property is a useful tool for the multiplication of integers. In the real world, we frequently use this property for mental multiplication. For example, -7(34) is easily done by completing -7(30) + -7(4) which can be represented by -7(30+4) = -210+-28=-238. Symbolically, the property is represented as: