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This essential trains for: SAT-I, GMAT, AMC-8, AMC-10, Math Kangaroo 3-4, Math Kangaroo 5-6, Math Kangaroo 7-8.

With respect to divisibility by 2, integers can be even or odd.

If the remainder of integer division by 2 is 0, then the integer is even. Using the theorem of integer division we can write such an integer N in a general way as a multiple of 2:

or simply:

If the remainder of integer division by 2 is 1, then the integer is odd. Using the theorem of integer division we can write such an integer N in a general way:

Obviously, the notions of odd and even refer only to integer division and have no meaning for numbers that are not integer.

Several easy but useful facts about the parity of integers:

Fact #1: Zero is even.

This is obvious since:

Fact #2: The sum/difference of two even integers is even:

Fact #3: The sum/difference of two odd integers is even:

Fact #4: The sum of an even and an odd number is odd.

Fact #5: If the sum of two arbitrary integers is even, so is their difference.

Subtract 2N from both sides:

The difference will result even:

Fact #6: If the difference of two arbitrary integers is even, so is their sum (proven similarly as the previous fact).

Fact #7: The product of two even numbers is even:

Fact #8: The product of two odd numbers is odd:

Fact #9: The product of an odd and an even is even:

Fact #10: If two numbers have an odd sum and an even difference then they cannot be integers. (this follows from facts #4 and #5 above).

Fact #11: If two numbers have an odd difference and an even sum then they cannot be integers. (this follows from facts #4 and #5 above).

Fact #12: No even number can ever be equal to an odd number. (The set of odd numbers and the set of even numbers are disjoint.)