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: Parity labels an integer even if it divides into two equal whole groups, odd if one is left over.

Common stuck point: The procedure for parity (even/odd) is the easy part; the trap is judging parity by size or digit count. Asking "Does the integer split into two equal whole groups with nothing left over?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does the integer split into two equal whole groups with nothing left over?

Worked Examples

Example 1

easy

Without fully computing, determine the parity (odd or even) of

2{,}345 + 6{,}782

and of

7 \times 14

Answer

2{,}345 + 6{,}782 = 9{,}127

is odd;

7 \times 14 = 98

is even.

First step

Parity rules: odd + even = odd.

2{,}345

is odd (ends in

5

);

6{,}782

is even (ends in

2

). Sum: odd.

Full solution

2
Parity rule for multiplication: odd $\times$ even = even. $7$ is odd; $14$ is even. Product: even.
3
Verify mentally: $2345 + 6782 = 9127$ (odd ✓); $7 \times 14 = 98$ (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.

Example 3

medium

Prove that the sum of two odd numbers is always even.

Example 4

medium

Is the sum of three consecutive integers always divisible by

3

? Use parity-style reasoning.

Example 5

medium

Prove that

n(n+1)

is always even for any integer

n

Example 6

hard

Prove that the square of any odd integer is odd.

Example 7

hard

In a chessboard tour, a knight alternates between dark and light squares. After

7

moves, can the knight return to its starting square?

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}

Example 3

easy

0

even or odd?

Example 4

easy

-7

even or odd?

Example 5

easy

What is the parity of

4+6

Example 6

easy

What is the parity of

3+8

Example 7

easy

What is the parity of

5\times 4

Example 8

easy

What is the parity of

3\times 7

Example 9

easy

48

even or odd?

Example 10

easy

How many odd numbers are there from

1

10

Example 11

medium

What is the parity of

7+7+7

Example 12

medium

Without computing, find the parity of

123+456

Example 13

medium

2^{10}

even or odd? Explain without full computation.

Example 14

medium

n

is odd, what is the parity of

n+1

? Of

2n

Example 15

medium

A number leaves remainder

1

when divided by

2

. What is its parity, and give an example over

20

Example 16

medium

What is the parity of the product of the first four counting numbers

1\times2\times3\times4

Example 17

medium

Two integers have an even sum. What can you say about their parities?

Example 18

medium

Without computing, find the parity of

246\times 357

Example 19

medium

a

and

b

are both odd, what is the parity of

a+b

and of

a\times b

Example 20

challenge

Prove that the product of two odd numbers is always odd.

Example 21

challenge

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

Example 22

challenge

In a room, people shake hands; each handshake involves two people. Use parity to show the number of people who shook an odd number of hands is even.

Example 23

easy

What is the parity of

9 + 11

Example 24

easy

What is the parity of

14 - 6

Example 25

easy

1{,}000

even or odd?

Example 26

easy

Is the sum

5 + 6 + 7

odd or even?

Example 27

medium

What is the parity of

7^3

Example 28

medium

What is the parity of

2^{10}

Example 29

medium

n

is odd, is

n^2 + n

odd or even?

Example 30

medium

What is the parity of

99 + 99

Example 31

medium

Two integers have the same parity. What is the parity of their sum?

Example 32

medium

Two integers have different parity. What is the parity of their sum?

Example 33

hard

What is the parity of the product

1 \cdot 2 \cdot 3 \cdots 20

Example 34

hard

How many odd integers are there from

1

100

inclusive?

Example 35

hard

What is the parity of

1 + 3 + 5 + \cdots + 99

Example 36

hard

Can the sum of three odd numbers equal

20

Example 37

hard

a + b

is odd and

a \cdot b

is even, what can you say about

a

and

b

Example 38

challenge

Twenty-five coins, all heads up, are on a table. Each move, you flip exactly two coins. Can you reach a state with exactly one tail?

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Related Concepts

Division Integers Divisibility Intuition

Background Knowledge

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

divisionintegers