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To prove that the difference of the squares of any two consecutive integers is odd, let be an integer number. The second of the consecutive integers is obtained by adding to Therefore, the consecutive integer to is Each number will be squared. Next, the difference of the squared numbers can be calculated. For simplicity the smaller integer is subtracted from the larger, but it should be noted that either way works. To determine whether this difference is an odd number, the expression can be divided by If the result is not an integer, the difference is odd. Since is an integer number and is not, the number is not an integer. This means that the number is odd for any integer Therefore, the difference of the squares of any two consecutive integers odd.

Any even number can be written as where is an integer. Consider the cube of this number. Now, when considering two consecutive integers, one is even and the other is odd. The odd number can be before or after These cases can be expressed by subtracting from or adding to respectively. Recall the formula for the cube of a binomial. By using this formula, the cubes of and can be found. These formulas are obtained by substituting for and for There are two possible situations. The consecutive numbers can be and or and These two situations will be considered one at a time.

and

If the consecutive integers are and then their cubes are and respectively. Next, find their difference. For the result to be positive, the cube of is subtracted from the cube of Now, divide this expression by An even number is always divisible by two. Therefore, if dividing the above expression by 2 results in an integer, the expression represents an even number. This expression will be examined closely to determine if it is an integer or not. Each term will be examined individually, gradually building the expression of

Expression	Is it an Integer?	Reason
	Yes	It is defined this way.
	Yes	Closure Property of Multiplication
	Yes	Closure Property of Multiplication
	Yes	Closure Property of Multiplication
	Yes	Closure Property of Subtraction
	No	is not an integer

Since dividing by did not result in an integer number, the expression which is the difference of the cubes of and is odd.

and

If the consecutive integers are and then their cubes are and respectively. Their difference can be calculated. Just like in the previous case, the expression that represents the difference will be divided by to determine if the result is even. Similar to the previous case, is an integer. Because of the the sum is not an integer. Therefore, the difference of the cubes is odd. Now that the two cases are considered, the statement has been proved.

Let and be two consecutive integer numbers. To prove the given statement, the numbers first must be added. The sum resulted in a binomial. Then, the square of the sum is the square of a binomial. An even number is always divisible by This means that, if dividing the above expression by results in an integer, the expression represents an even number. Otherwise, it represents an odd number. Since is an integer, the expression also represents an integer. Adding a non-integer to an integer results in a non-integer. Therefore, the expression is not an integer. This means that is an odd number, proving the statement.