Watch out Properties of integers with Mathematical Division

Mathematical Division of integers is neither commutative nor associative. 
Mathematical Division of integers can be represented as following

a / b != b / a

(a / b) / c != a / (b / c)

Justifying commutative property of mathematical Division of integers

a / b != b / a

If a = 4 and b = 2

Then 4 / 2 = 2 and 2 / 4 = 0.5

So, 4 / 2 != 2 / 4 as 2 != 0.5

Justifying associative property of mathematical Division of integers

(a / b) / c != a / (b / c)

If a = 4, b = 2 and c = 8

Then (4 / 2) / 8 = 0.25 and 4 / (2 / 8) = 16

So, (4 / 2) / 8 != 4 / (2 / 8) as 0.25 != 16

Watch out Properties of integers with Mathematical Subtraction

Mathematical subtraction of integers is neither commutative nor associative. 
Mathematical subtraction of integers can be represented as following

a - b != b - a

(a - b) - c != a - (b - c)

Justifying commutative property of mathematical subtraction of integers

a - b != b - a

If a = 5 and b = 3

Then 5 - 3 = 2 and 3 - 5 = (-2)

So, 5 - 3 != 3 - 5 as 2 != (-2)

Justifying associative property of mathematical subtraction of integers

(a - b) - c != a - (b - c)

If a = 5, b = 3 and c = 7

Then (5 - 3) - 7 = (-5) and 5 - (3 - 7) = 9

So, (5 - 3) - 7 != 5 - (3 - 7) as (-5) != 9

Properties of integers

Useful properties of integers to solve math problems are as follows ... 

1. Commutative property of mathematical addition of integers.
2. Commutative property of mathematical multiplication of integers.
3. Associative property of mathematical addition of integers.
4. Associative property of mathematical multiplication of integers.
   
5. The Distributive property of integers.

The Distributive property of integers

The Distributive property of integers affects when multiplication and summation operations are performed together. 
The Distributive property of integers indicates distribution of operations means the answer of multiplication operation performed after the summations and the answer of multiplication operation performed before the summations are equal. 
The Distributive property of integers can be represented as following

a (b + c) = ab + ac

OR

a x (b + c) = (a x b) + (a x c)

For example,

If a = 5, b = 3 and c = 7

Then 5 x (3 + 7) = 50 and (5 x 3) + (5 x 7) = 50

So, 5 x (3 + 7) = (5 x 3) + (5 x 7) as 50 = 50

Another example,

If a = 5, b = -3 and c = 7

Then 5 x ((-3) + 7) = 20 and (5 x (-3)) + (5 x 7) = 20

So, 5 x ((-3) + 7) = (5 x (-3)) + (5 x 7) as 20 = 20

Associative property of mathematical multiplication of integers

The associative property of mathematical multiplication of integers says that any order of grouping of numbers in multiplication results into same answer. 
The associative property of mathematical multiplication of integers can be represented as following

(ab)c = a(bc)

OR

(a x b) x c = a x (b x c)

To understand associative property of mathematical multiplication of integers we can try multiplying numbers in different orders and note the results. 
For example,

If a = 5, b = 3 and c = 7

Then (5 x 3) x 7 = 105 and 5 x (3 x 7) = 105

So, (5 x 3) x 7 = 5 x (3 x 7) as 105 = 105

 
Another example,

If a = 5, b = -3 and c = 7

Then (5 x (-3)) x 7 = -105 and 5 x ((-3) x 7) = -105

So, (5 x (-3)) x 7 = 5 x ((-3) x 7) as -105 = -105

Associative property of mathematical addition of integers

The associative property of mathematical addition of integers says that any order of grouping of numbers in addition/summation results into same answer. 
The associative property of mathematical addition of integers can be represented as following

(a + b) + c = a + (b + c)

To understand associative property of mathematical addition of integers we can try adding numbers in different orders and note the results. 
For example,

If a = 5, b = 3 and c = 7

Then (5 + 3) + 7 = 15 and 5 + (3 + 7) = 15

So, (5 + 3) + 7 = 5 + (3 + 7) as 15 = 15

 
Another example,

If a = 5, b = -3 and c = 7

Then (5 + (-3)) + 7 = 9 and 5 + ((-3) + 7) = 9

So, (5 + (-3)) + 7 = 5 + ((-3) + 7) as 9 = 9