Parity is the mathematical word for whether a number is even or odd. If a number is even, its parity is even, and if it is odd, its parity is odd.
What is Parity?
Parity tells us if a number is even or odd:
- An even number leaves no remainder when divided by \(2\). It is called as even parity (eg: \(6 \div 2 = 3\) with remainder \(0\)).
- An odd number leaves a remainder of \(1\) when divided by \(2\). It is called as odd parity (eg: \(7 \div 2 = 3\) with remainder \(1\)).
Addition and Subtraction Parity:
odd \(\pm\) odd \(=\) even
even \(\pm\) even \(=\) even
odd \(\pm\) even \(=\) odd
Product of Parity:
odd \(\times\) even \(=\) even
odd \(\times\) odd \(=\) odd
Exponent Parity:
odd \(^3 =\) odd
Important!
Even number = any integer of the form \(2n\)
Odd number = any integer of the form \(2n + 1\)
Breaking Even:
We can check if arithmetic or algebraic expressions always give even numbers without actually calculating.
| Arithmetic Expression | Parity Condition | Result |
| \(65+29\) | odd \(+\) odd \(=\) even | Even |
| \(386 - 214\) |
even \(-\) even \(=\) even
|
Even |
| \(235+46\) |
odd \(+\) even \(=\) odd
|
Odd |
| \(8 \times 367\) | odd \(\times\) even \(=\) even | Even (\(8\) is Even) |
| \(315^3\) | odd \(^3 =\) odd | Odd |
Let us recall the factors and multiples before learning further:
A factor of a number divides it exactly without leaving any remainder.
Example:
The factors of \(18\) are:
\(1\), \(2\), \(3\), \(6\), \(9\) and \(18\)
Since \(18 ÷ 3 = 6\)
\(3\) is a factor of \(18\).
A multiple is obtained by multiplying a number by a whole number.
Example:
Multiples of \(6\)
\(6\), \(12\), \(18\), \(24\), \(30\), \(36\), ...
Every multiple of \(6\) is obtained by multiplying \(6\) by \(1\), \(2\), \(3\), \(4\), ...
We learned about differentiating even and odd numbers of arithmetic expressions. Let us expand the same to algebraic expressions.
| Algebraic Expression | Can \(2\) be taken as a common factor? | Always Even? | Why? | Example |
|---|---|---|---|---|
| \(2p + 6q\) | Yes | ✅ Always | Both terms have a factor of \(2\). So, (\(2p + 6q = 2(p + 3q)\)). | If (\(p=2,\ q=1\)): (\(4+6=10\)) (Even) |
| \(4m − 8n\) | Yes | ✅ Always | Both terms are multiples of \(2\). The expression can be written as (\(4(m-2n)\)), so it is always even. | If (\(m=5,\ n=2\)): (\(20-16=4\)) (Even) |
| \(10x + 14y\) | Yes | ✅ Always | Each term has a factor of \(2\). Therefore, the entire expression is always divisible by \(2\). | If (\(x=1,\ y=2\)): (\(10+28=38\)) (Even) |
|
\(7a + 9b\)
|
No | ❌ Sometimes | There is no common factor of \(2\). The result depends on whether (\(a\)) and (\(b\)) are even or odd. | (\(a=1,\ b=1\)): (\(16\)) (Even); (\(a=2,\ b=1\)): (\(23\)) (Odd) |
| \(5x − 3y\) | No | ❌ Sometimes | Since \(2\) is not a common factor, the parity changes with different values of (\(x\)) and (\(y\)). | (\(x=3,\ y=1\)): (\(12\)) (Even); (\(x=2,\ y=1\)): (\(7\)) (Odd) |
| \(n² + 2\) | No | ❌ Sometimes | If (\(n\)) is even, the result is even. If (\(n\)) is odd, the result becomes odd. | (\(n=4\)): (\(18\)) (Even); (\(n=3\)): (\(11\)) (Odd) |
| \(m² + 1\) | No | ❌ Sometimes | The parity depends on whether (\(m\)) is even or odd. | (\(m=4\)): (\(17\)) (Odd); (\(m=5\)): (\(26\)) (Even) |
| \(6r × 5s\) | Yes | ✅ Always | Since \(6\) is even, the product always contains a factor of \(2\). | (\(r=2,\ s=3\)): (\(12×15=180\)) (Even) |
| \(12a − 4b\) | Yes | ✅ Always | Both terms are divisible by \(4\) (and hence by \(2\)). Therefore, the result is always even. | (\(a=3,\ b=5\)\): (\(36-20=16\)) (Even) |
For the expressions that are sometimes even, \(2\) cannot be taken as a common factor from every term. Their result depends on the values of the variables.
Important!
An algebraic expression is always even if every term has a factor of \(2\).
In other words, if the expression can be written as
\(2×\)(another whole-number expression),
then the expression is always even.
Example: \(8a+10b =2(4a+5b)\)
Since \(2\) is a common factor, the expression is always even, no matter what values \(a\) and \( take.