Parity (Even/Odd) Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Parity (Even/Odd).

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The classification of integers as even (evenly divisible by 2, with no remainder) or odd (not divisible by 2).

Can you split it into two equal groups? Yes = even, no = odd.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Even + even = even. Odd + odd = even. Even + odd = odd.

Common stuck point: Zero is even (0 = 2 \times 0). Negative numbers have parity too.

Sense of Study hint: Divide the number by 2. If the result is a whole number with no remainder, it is even. If there is a remainder of 1, it is odd.

Worked Examples

Example 1

easy
Without fully computing, determine the parity (odd or even) of 2{,}345 + 6{,}782 and of 7 \times 14.

Solution

  1. 1
    Parity rules: odd + even = odd. 2{,}345 is odd (ends in 5); 6{,}782 is even (ends in 2). Sum: odd.
  2. 2
    Parity rule for multiplication: odd \times even = even. 7 is odd; 14 is even. Product: even.
  3. 3
    Verify mentally: 2345 + 6782 = 9127 (odd ✓); 7 \times 14 = 98 (even ✓).

Answer

2{,}345 + 6{,}782 = 9{,}127 is odd; 7 \times 14 = 98 is even.
Parity (odd/even) is determined by the last digit and follows simple rules: odd + even = odd; even + even = even; odd + odd = even; any number times an even is even; odd \times odd = odd. These rules let us classify results without computing.

Example 2

medium
Prove that the sum of any two consecutive integers is always odd.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Classify each as odd or even without fully computing: (a) 100 + 201, (b) 6 \times 7 \times 8, (c) 15^2.

Example 2

medium
In a group of 50 people, each person shakes hands with every other person exactly once. Is the total number of handshakes odd or even? (Hint: use the formula \frac{n(n-1)}{2}).
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Background Knowledge

These ideas may be useful before you work through the harder examples.

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