An integer is even if it is evenly divisible by 2.^{[3]} Expressed mathematically:
"Let n be an integer. We say that n is even if there is some integer k such that n = 2k."^{[4]}In set notation, even integers are .
In the decimal numbering system, an integer can be identified as even by the fact that the last digit of the number is even. Since the digits 0, 2, 4, 6 and 8 are even, the numbers 2750, -54, 22, -888 and 1794830495907549234098546 are even. Any integer in decimal form that does not have an even digit as the last digit is odd.
Whether an integer is even or odd is called parity. One says, "The parity of 6 is even," or "The parity of -365 is odd."
Consider two arbitrary even integers x and y. | Initial assertion |
An even integer can be written as 2a where a is an integer. | Definition of an even integer. |
We will show that there exists an integer c such that x + y = 2c. | Claim |
x and y can be rewritten as x = 2a and y = 2b where a and b are integers. | Apply the definition of an even integer. |
The sum x + y can be written as 2a + 2b. | Substitute 2a for x and 2b for y. |
2a + 2b can be written as 2(a + b). | Apply the distributive property of multiplication. |
There exists an integer c = a + b. | Apply the closure property of integers and addition. |
2(a + b) can then be rewritten as 2(c) = 2c. | Substitute c for a + . |
QED. | The proof is complete. |
Consider two arbitrary even integers x and y. | Initial assertion |
An even integer can be written as 2a where a is an integer. | Definition of an even integer. |
We will show that there exists an integer c such that x · y = 2c. | Claim |
x and y can be rewritten as x = 2 and y = 2b where a and b are integers. | Apply the definition of an even integer. |
The product x · y can be written as 2a · 2b. | Substitute 2a for x and 2b for y. |
2a · 2 can be written as 2 · (a · 2 · b). | Apply the associative property of multiplication. |
There exists an integer c = a · 2 · b. | Apply the closure property of integers and multiplication. |
2(a · 2 · ) can then be rewritten as 2(c) = 2c. | Substitute c for a · b. |
QED. | The proof is complete. |
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