Even and Odd Numbers
Every whole number is either even or odd. This is one of the simplest and most useful ways to classify numbers.
When you share sweets equally between two friends, if nothing is left over, the total is even. If one sweet is left over, the total is odd.
Even numbers are exactly divisible by 2: 0, 2, 4, 6, 8, 10, 12, ... They can be split into two equal groups.
Odd numbers are not exactly divisible by 2: 1, 3, 5, 7, 9, 11, 13, ... When split into two groups, there is always 1 left over.
What is Even and Odd Numbers?
Definition:
- An even number is a whole number that is exactly divisible by 2 (leaves remainder 0 when divided by 2).
- An odd number is a whole number that is not exactly divisible by 2 (leaves remainder 1 when divided by 2).
Even numbers: 0, 2, 4, 6, 8, 10, 12, ...
Odd numbers: 1, 3, 5, 7, 9, 11, 13, ...
Quick identification:
- Look at the last digit (ones place) of the number.
- If it ends in 0, 2, 4, 6, or 8 → the number is even.
- If it ends in 1, 3, 5, 7, or 9 → the number is odd.
Using algebra:
- Even numbers can be written as 2n (where n is a whole number). Examples: 2×0=0, 2×1=2, 2×2=4, 2×3=6.
- Odd numbers can be written as 2n + 1. Examples: 2×0+1=1, 2×1+1=3, 2×2+1=5, 2×3+1=7.
Special case:
- 0 is even because 0 ÷ 2 = 0 with no remainder.
- 0 = 2 × 0, so it follows the form 2n.
Even and Odd Numbers Formula
Rules for Operations with Even and Odd Numbers:
Addition:
Even + Even = Even
Odd + Odd = Even
Even + Odd = Odd
Subtraction:
Even − Even = Even
Odd − Odd = Even
Even − Odd = Odd
Odd − Even = Odd
Multiplication:
Even × Even = Even
Odd × Odd = Odd
Even × Odd = Even
Key insight: Multiplying by any even number always gives an even result. The only way to get an odd product is when both factors are odd.
Types and Properties
Types of even numbers:
- Positive even numbers: 2, 4, 6, 8, 10, ...
- 0 is even (but neither positive nor negative).
- Negative even numbers: -2, -4, -6, -8, ... (for integers).
- The smallest positive even number is 2.
Types of odd numbers:
- Positive odd numbers: 1, 3, 5, 7, 9, ...
- Negative odd numbers: -1, -3, -5, -7, ... (for integers).
- The smallest positive odd number is 1.
Even and odd in number sequences:
- Even and odd numbers alternate on the number line: 0 (E), 1 (O), 2 (E), 3 (O), 4 (E), 5 (O), ...
- The predecessor and successor of every even number are odd.
- The predecessor and successor of every odd number are even.
Interesting facts:
- 2 is the only even prime number. All other even numbers are composite (divisible by 2).
- The sum of any two consecutive numbers (one even, one odd) is always odd.
- The product of any two consecutive numbers is always even.
Solved Examples
Example 1: Identify Even or Odd
Problem: Classify each as even or odd: 38, 127, 540, 3,005, 91.
Solution:
Steps:
- Check the last digit of each number.
- 38 → last digit 8 → Even.
- 127 → last digit 7 → Odd.
- 540 → last digit 0 → Even.
- 3,005 → last digit 5 → Odd.
- 91 → last digit 1 → Odd.
Answer: Even: 38, 540. Odd: 127, 3005, 91.
Example 2: Even + Even
Problem: Is the sum 24 + 36 even or odd? Verify.
Solution:
Given:
- 24 (even) + 36 (even)
Steps:
- Rule: Even + Even = Even.
- Verify: 24 + 36 = 60. Last digit = 0 → Even.
Answer: 24 + 36 = 60 (Even).
Example 3: Odd + Odd
Problem: Is the sum 15 + 27 even or odd?
Solution:
Given:
- 15 (odd) + 27 (odd)
Steps:
- Rule: Odd + Odd = Even.
- Verify: 15 + 27 = 42. Last digit = 2 → Even.
Answer: 15 + 27 = 42 (Even).
Example 4: Even + Odd
Problem: Is the sum 18 + 7 even or odd?
Solution:
Given:
- 18 (even) + 7 (odd)
Steps:
- Rule: Even + Odd = Odd.
- Verify: 18 + 7 = 25. Last digit = 5 → Odd.
Answer: 18 + 7 = 25 (Odd).
Example 5: Multiplication Rules
Problem: Without calculating, predict whether 13 × 24 is even or odd.
Solution:
Given:
- 13 (odd) × 24 (even)
Steps:
- Rule: Even × Odd = Even.
- Whenever one factor is even, the product is even.
Answer: 13 × 24 is Even. (Actual: 312, which ends in 2.)
Example 6: Odd × Odd
Problem: Without calculating, is 7 × 9 even or odd?
Solution:
Given:
- 7 (odd) × 9 (odd)
Steps:
- Rule: Odd × Odd = Odd.
Answer: 7 × 9 is Odd. (Actual: 63, which ends in 3.)
Example 7: Finding Even Numbers in a Range
Problem: How many even numbers are there between 1 and 50 (not including 1 and 50)?
Solution:
Steps:
- Even numbers between 1 and 50 (exclusive): 2, 4, 6, ..., 48.
- This is an arithmetic sequence with first term 2, last term 48, common difference 2.
- Count = (48 − 2) / 2 + 1 = 46/2 + 1 = 23 + 1 = 24.
Answer: There are 24 even numbers between 1 and 50.
Example 8: Sum of First 5 Even Numbers
Problem: Find the sum of the first 5 even numbers (starting from 2).
Solution:
Given:
- First 5 even numbers: 2, 4, 6, 8, 10
Steps:
- Sum = 2 + 4 + 6 + 8 + 10 = 30.
- Formula: Sum of first n even numbers = n(n + 1) = 5 × 6 = 30.
Answer: Sum = 30.
Example 9: Sum of First 5 Odd Numbers
Problem: Find the sum of the first 5 odd numbers.
Solution:
Given:
- First 5 odd numbers: 1, 3, 5, 7, 9
Steps:
- Sum = 1 + 3 + 5 + 7 + 9 = 25.
- Pattern: Sum of first n odd numbers = n² = 5² = 25.
Answer: Sum = 25.
Example 10: Is 0 Even or Odd?
Problem: Determine whether 0 is even or odd.
Solution:
Steps:
- A number is even if it is divisible by 2 with no remainder.
- 0 ÷ 2 = 0 with remainder 0.
- Also, 0 = 2 × 0, so it fits the form 2n.
Answer: 0 is even.
Real-World Applications
Where even and odd numbers are used:
- Sharing equally: If you have an even number of items, you can share them equally between 2 people with nothing left over.
- Pairing: In PE class, if the number of students is even, everyone gets a partner. If odd, one student is left without a partner.
- House numbers: On many streets, even-numbered houses are on one side and odd-numbered houses are on the other side.
- Games: "Odds and Evens" is a hand game similar to rock-paper-scissors. Players show fingers and check if the total is odd or even.
- Computer science: Computers use even/odd checks (parity bits) to detect errors in data transmission.
- Calendars: Even and odd years alternate. Even months (Feb, Apr, Jun, etc.) and odd months have different numbers of days.
Key Points to Remember
- A number is even if it ends in 0, 2, 4, 6, or 8. A number is odd if it ends in 1, 3, 5, 7, or 9.
- Even numbers are divisible by 2 with no remainder. Odd numbers leave remainder 1.
- 0 is even.
- Even + Even = Even. Odd + Odd = Even. Even + Odd = Odd.
- Even × anything = Even. Odd × Odd = Odd.
- 2 is the only even prime number.
- Even and odd numbers alternate on the number line.
- Sum of first n even numbers = n(n + 1).
- Sum of first n odd numbers = n².
- Consecutive numbers are always one even and one odd.
Practice Problems
- Classify as even or odd: 256, 1,333, 70, 49, 1,000.
- Without calculating, is 45 + 32 even or odd?
- Without calculating, is 11 × 15 even or odd?
- List all odd numbers between 40 and 60.
- Find the sum of the first 8 even numbers.
- If the product of two numbers is odd, what can you say about the two numbers?
- Is the number 2 × 3 × 5 × 7 even or odd? Explain.
- A class has 37 students. Can they be divided into groups of 2 with no one left out?
Frequently Asked Questions
Q1. How do you tell if a number is even or odd?
Look at the last digit (ones place). If it is 0, 2, 4, 6, or 8, the number is even. If it is 1, 3, 5, 7, or 9, the number is odd.
Q2. Is 0 even or odd?
0 is even. It is divisible by 2 (0 ÷ 2 = 0 with remainder 0) and can be written as 2 × 0.
Q3. What happens when you add two odd numbers?
The sum of two odd numbers is always even. Example: 3 + 7 = 10 (even). 15 + 9 = 24 (even).
Q4. Can the product of an even and an odd number be odd?
No. If any factor is even, the product is always even. The only way to get an odd product is when all factors are odd.
Q5. Why is 2 the only even prime?
A prime number has exactly two factors: 1 and itself. 2 has factors 1 and 2 — it is prime. Every other even number is divisible by 2 AND another number, so it has more than 2 factors and is composite.
Q6. What is the sum of the first n odd numbers?
The sum of the first n odd numbers equals n². For example: 1+3+5 = 9 = 3². 1+3+5+7 = 16 = 4².
Q7. Are negative numbers even or odd?
Yes, negative numbers are also classified as even or odd. -2, -4, -6 are even. -1, -3, -5 are odd. The rule is the same: check divisibility by 2.
Q8. How many even numbers are between 1 and 100?
The even numbers from 2 to 100 are: 2, 4, 6, ..., 100. There are 50 even numbers. Formula: (100 - 2)/2 + 1 = 50.
