4. Odd and Even Numbers

Introduction to Odd and Even Numbers

Definition:

• Even numbers: Whole numbers that are divisible by 2 without a remainder. Examples: 0,2,4,6,8,…
• Odd numbers: Whole numbers that are not divisible by 2. Examples: 1,3,5,7,9,…

Key Properties of Even and Odd Numbers:

1. The last digit of an even number is 0,2,4,6,or 8
2. The last digit of an odd number is 1,3,5,7,or 9
3. Sum/Difference of Two Numbers:
   • Even +/− Even = Even.
   • Odd +/− Odd = Even.
   • Even +/− Odd = Odd.
4. Product of Two Numbers:
   • Even × Even = Even.
   • Odd × Odd = Odd.
   • Even × Odd = Even.

Basic Examples

1. Example 1: Is 245 odd or even?
   Solution: Look at the last digit (5), which is odd. Hence, 245 is an odd number.
2. Example 2: Add 14 (even) and 27 (odd).
   Solution: 14 + 27 = 41 (odd)
3. Example 3: Multiply 8 (even) and 9 (odd).
   Solution: 8 × 9 = 72 (even)

Intermediate Concepts

1. Checking Large Numbers:
For any number, focus only on the last digit to determine if it is odd or even.

• Example: 2457682 → Last digit is 2. Therefore, 2457682 is even.

2. Sequences of Odd and Even Numbers:
Odd and even numbers alternate in a sequence.

• Example: 1,2,3,4,5,6,…

3. Division by 2:

• If the quotient of a number divided by 2 is an integer, the number is even.
• If not, the number is odd.
  • Example: 45 ÷ 2 = 22.5 → 45 is odd.
  • Example: 88 ÷ 2 = 44 → 88 is even.

Advanced Concepts

1. Difference of Consecutive Even/Odd Numbers:
The difference between two consecutive even or odd numbers is always $2$.

• Example: $$12 – 10 = 2, \quad 19 – 17 = 2.$$

2. Product of Consecutive Numbers:
The product of two consecutive integers is always even (one number is always even).

• Example: $$6\times 7=42\text{(even)}\mathrm{.}$$

 

3. Sum of  first n consecutive Even Numbers:
      = n ( n + 1 )

• Example: Sum of the first 5  even numbers:

             2 + 4 + .. + 2 x 5

             2 + 4 + .. + 10= 5 ( 5 + 1) = 30

• Example: Sum of the first 50  even numbers:

             2 + 4 + 6 + 8 + 10 … + 2 x 50  = 50( 50 + 1) =  2550

4. Sum of first n consecutive Odd Numbers:
The sum of the first $n$ odd numbers is a perfect square:

$$1 + 3 + 5 + \dots + (2n – 1) = n^2.$$

• Example: Sum of the first $5$ odd numbers: $$1 + 3 + 5 + 7 + 9 = 5^2 = 25.$$
  

5. Sum of n Numbers in a series Or Sum of Consecutive n Numbers: :

In a number series difference between any 2 consecutive numbers is same.

Examples of number series 

• 1, 2, 3, 4, 5, 6
• 2, 4, 6, 8, 10 ( series of first n Even numbers )
• 1, 3, 5, 7, 9 ( series of first n Odd numbers )
• 10, 20, 30, 40, 60
• 10, 14, 18, 22, 26

The sum of the $n$ consecutive numbers or series of n numbers = \(\Large  \mathtt{\frac{( first\ number\ +\ last\ number) \times \ n}{2}} \) 

• Example: Sum of the first 5  natural numbers:

          1+2+3+4+5 = \(\Large  \mathtt{\frac{( 1 + 5) \times \ 5}{2}} \)

                                = \(\Large  \mathtt{\frac{ 6 \times \ 5}{2}} \)

                                = 15

• Example: Sum of the consecutive numbers from 10 to 20:

      10+11+12+….+20 = \(\Large  \mathtt{\frac{( 10 + 20) \times \ 11 }{2}} \)

                                = \(\Large  \mathtt{\frac{ 30 \times \ 11}{2}} \)

                                = 165

6 . Finding Nth term in the number series or count of numbers in series: :

 Assume :

     first term in series = f

     count of terms in series i.e. count of numbers in series = n

    difference between 2 terms in series = d   

Nth term in a series = l

Then –

               l = f + (n-1) x d

Last term =

First term + ( Count of terms in the series – 1 ) x Difference between 2 consecutive terms of series

 

Very Advanced Concepts

1. Modular Arithmetic with Odd and Even Numbers:

• Even numbers can be written as $2k$ (where k is an integer).
• Odd numbers can be written as $\mathrm{2k}+1$ (where k is an integer).
• Operations:
  • Even + Even $\equiv 0 (\text{mod } 2)$
  • Odd + Odd $\equiv 0 (\text{mod } 2)$
  • Even + Odd $\equiv 1 (\text{mod } 2)$

 

Practice Questions with Solutions

 

1. Determine if $739$ is odd or even.
Solution: Last digit is $9$, so $739$ is odd.

2. What is the sum of all odd numbers between $1$ and $15$?
Solution:

$$1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 = 64.$$

= $$1+3+5+7+9+11+13+\; (2\; x\; 8\; -1\; )$$

= \( \large 8^2 \)

=  64

3.  The product of any three consecutive integers is always even or odd  ?
Solution: In a series of 3 consecutive numbers, one number is always even, so the product is even.

4.  Identify if $246801$ is odd or even.
Solution: Last digit is $1$, so $246801$ is odd.

5. What is the sum of $99$ odd numbers starting from $1$?
Solution:

$$\text{Sum} = 99^2 = 9801.$$

6. Determine the smallest even number divisible by $\mathrm{}\text{ 3 and 5.}$

Solution: LCM of $3\mathrm{and\; 5}=15$

but 15 is not even number, so we need to find the smallest multiple of 15 which is even.

Thus smallest even number divisible 3 and 5 = 15 x 2 = 30$3, 4, \text{and } 6 = 12$

7. Is the sum of $23, 34, \text{and } 41$ odd or even?
Solution:

$$23 + 34 + 41 = 98 \quad \text{(even)}.$$

8. Find the largest odd factor of $120$.
Solution:
Prime factorisation: $120 = 2^3 \times 3 \times 5$.

Largest odd factor: $3 \times 5 = 15$

9. Can the product of $10$ with odd numbers be even?
Solution: Yes, the product of even number with odd numbers is always even.

10. If $x$ is even, prove that $x^2$ is even.
Solution: Square of any even number is even because $2k \times 2k = 4k^2$

11. Is the sum of two even numbers always even?
Solution: Yes, $2m + 2n = 2(m+n)$ which is even.

12. Calculate the sum of first $50$ even numbers.
Solution:

$$\text{Sum} = 2(1 + 2 + \dots + 50) = 2 \times \frac{50 \times 51}{2} = 2550.$$

13. Determine the smallest odd number greater than $1000$.
Solution: $1001$.

14. Prove that $2^{n+1}$ is always even.

Solution: $2^{n+1} = 2 \times 2^n$  which is divisible by $2$.

15. Find the total number of even numbers between $1$ and $200$.
Solution:
Even numbers between $1$ and $200$ form an arithmetic sequence:

$$2, 4, 6, \dots, 200.$$

The $n$-th term of an arithmetic sequence is given by:

$$a_n = a + (n-1)d.$$Here, $a = 2$, $d = 2$, and $a_n = 200$, Solve for $n$:

$$200 = 2 + (n-1) \times 2 \quad \Rightarrow \quad 200 = 2n \quad \Rightarrow \quad n = 100.$$Answer: There are $100$ even numbers between $1$ and $200$.

16. Prove that the product of two consecutive even numbers is always divisible by 8$8$.
Solution:
Let the two consecutive even numbers be $n = 2k$ and $n+2 = 2k+2$.
The product is:

$$n \times (n+2) = 2k \times (2k+2) = 4k(k+1).$$

• One of $k$ or $k+1$ is always even, making $4k(k+1)$ divisible by $8$.

Answer: The product is always divisible by $8$.

 

17. Find the sum of all even numbers between $1$ and $100$.
Solution:
Even numbers between 1 and $100$ form an arithmetic sequence:

$$2, 4, 6, \dots, 100.$$

Number of terms ($n$):

$$100 = 2 + (n-1) \times 2 \quad \Rightarrow \quad n = 50.$$Sum of an arithmetic sequence:

$$S_n = \frac{n}{2} \times (a + l).$$Here, $n = 50$, $a = 2$, $l = 100$:

$$S_n = \frac{50}{2} \times (2 + 100) = 25 \times 102 = 2550.$$

Answer: The sum is $2550$.

 

18. Can the sum of $7$ odd numbers be even?
Solution:
The sum of odd numbers follows this rule:

• Odd + Odd = Even
• Adding another odd number to an even sum makes it odd.

Thus, the sum of $7$ odd numbers is always odd.

 

19. If 𝑥 is an odd number,  prove that  $x^2 – 1$ is always divisible by $8$.
Solution:
Let $x = 2k+1$ (general form of an odd number).

$$x^2 – 1 = (2k+1)^2 – 1 = 4k^2 + 4k + 1 – 1 = 4k(k+1).$$

• $k(k+1)$ is always even (product of two consecutive integers).
• Therefore, $4k(k+1)$ is divisible by $8$.

Answer: $x^2 – 1$ is always divisible by $8$.

 

20. Find all even numbers divisible by both $3$ and $4$ between $1$ and $100$.
Solution:
An even number divisible by $3$ and $4$ must also be divisible by their LCM.

$$\text{LCM of } 3 \text{ and } 4 = 12.$$

Even numbers divisible by $12$:

$$12, 24, 36, 48, 60, 72, 84, 96.$$

Count of numbers: $8$.

Answer: The numbers are $12, 24, 36, 48, 60, 72, 84, 96$

 

More Practice Questions

 

1. What is the sum of all odd numbers between $1$ and $99$?

2. If $x$ is an even number, prove that $x^2 + 3x$ is always even.

3. How many odd numbers between $1$ and $1000$ are divisible by $5$?

4. Prove that the difference between the squares of two consecutive odd numbers is always divisible by $8$.

5. If $n$ is odd, prove that $n^3 – n$ is divisible by $8$.

6. Find the sum of all even numbers between $1$ and $200$ that are divisible by $5$.

7. Find the smallest even number greater than $100$ that is divisible by $7$.

8. If $x$ is odd,  $x^2 – 1$ is divisible by $4$ or not ?

9. Determine count of all even numbers between $1$ and $500$ that are divisible by both $6$ and $8$.

10.  Product of three consecutive odd numbers is odd or even ?

11. How many odd numbers between $1$ and $300$ are divisible by $9$?

12. If $x$ is an even number and $y$ is an odd number,  $x^2 – y^2$ is always even or odd ?

13. Find the sum of all odd numbers between $200$ and $400$.

14. How many even numbers between $1$ and $1000$ are divisible by both $4$ and $5$?

15.  The sum of any $n$ consecutive even numbers is divisible by n or not ?.

16. How many odd numbers between $1$ and $500$ are divisible by both $3$ and $7$?

17. Find the smallest odd number greater than $1000$ that is divisible by $11$.

18.  The product of any three consecutive even numbers is always divisible by $48$ ?

19. If $n$ is an odd number, prove that $n^3 + n$ is always divisible by $4$.

20. Find all even numbers between $1$ and $200$ that are divisible by $3$ but not by $9$.