Properties of Numbers

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Even and Odd Numbers

Definition

Numbers can be classified as even or odd based on their divisibility by 2.

  • Even Numbers: Numbers that are divisible by 2 (i.e., leave no remainder when divided by 2).
    Example: 2, 4, 6, 8, 10, …
    Mathematical form: n=2kn=2k, where kk is an integer.
  • Odd Numbers: Numbers that are not divisible by 2 (i.e., leave a remainder of 1 when divided by 2).
    Example: 1, 3, 5, 7, 9, …
    Mathematical form: n=2k+1n=2k+1, where kk is an integer.

Properties of Even and Odd Numbers

  1. Even + Even = Even
    Example: 4 + 6 = 10
  2. Odd + Odd = Even
    Example: 3 + 5 = 8
  3. Even + Odd = Odd
    Example: 4 + 7 = 11
  4. Even × Even = Even
    Example: 2 × 6 = 12
  5. Odd × Odd = Odd
    Example: 3 × 5 = 15
  6. Even × Odd = Even
    Example: 4 × 5 = 20

Prime and Composite Numbers

Prime Numbers

A prime number is a number that has exactly two factors: 1 and itself.

Example: 2, 3, 5, 7, 11, 13, 17, 19, …

Important Properties:

  • 2 is the only even prime number.
  • All other even numbers greater than 2 are composite.
  • Prime numbers greater than 2 are always odd.

Composite Numbers

A composite number has more than two factors (i.e., it can be divided by numbers other than 1 and itself).

Example: 4, 6, 8, 9, 10, 12, 15, 18, …

Example of Factorization:

  • 6: Factors = 1, 2, 3, 6 (Composite)
  • 13: Factors = 1, 13 (Prime)
  • 15: Factors = 1, 3, 5, 15 (Composite)

Special Cases:

  • 1 is neither prime nor composite because it has only one factor (1 itself).

Co-prime Numbers (Relatively Prime Numbers)

Definition

Two numbers are co-prime if they do not have any common factor other than 1.

This means their HCF (Highest Common Factor) is 1.

Example:

  • (5, 9) → Co-prime (Factors of 5 = 1, 5; Factors of 9 = 1, 3, 9 → Only common factor is 1)
  • (8, 12) → Not Co-prime (Common factors: 1, 2, 4)

Properties of Co-prime Numbers

  1. Two consecutive numbers are always co-prime.
    Example: (7, 8), (14, 15)
  2. The LCM of two co-prime numbers is always their product.
    Example: LCM of (3, 7) = 3 × 7 = 21
  3. The HCF of two co-prime numbers is always 1.

Divisibility Rules

Divisibility rules help in quickly determining whether a number is divisible by another number without actual division.

Divisibility Rules from 2 to 11

NumberRuleExample
2A number is divisible by 2 if it ends in 0, 2, 4, 6, or 8.38 is divisible by 2 because it ends in 8.
3A number is divisible by 3 if the sum of its digits is a multiple of 3.234 → 2+3+4 = 9 (divisible by 3).
4A number is divisible by 4 if the last two digits form a number that is divisible by 4.316 → 16 is divisible by 4.
5A number is divisible by 5 if it ends in 0 or 5.145, 250 (Both end in 5 or 0).
6A number is divisible by 6 if it is divisible by both 2 and 3.48 → Ends in 8 (divisible by 2), 4+8=12 (divisible by 3) → Divisible by 6.
7Double the last digit, subtract it from the rest of the number, and check if the result is divisible by 7.203 → 20 - (3×2) = 20 - 6 = 14(divisible by 7).
8A number is divisible by 8 if the last three digits form a number that is divisible by 8.1024 → 024 is divisible by 8.
9A number is divisible by 9 if the sum of its digits is a multiple of 9.729 → 7+2+9 = 18 (divisible by 9).
10A number is divisible by 10 if it ends in 0.250, 780 (Both end in 0).
11A number is divisible by 11 if the difference between the sum of odd-positioned digits and even-positioned digits is a multiple of 11.121 → (1+1) - (2) = 0 (divisible by 11).

Practice MCQs on These Topics

1. Which of the following is an odd number?

a) 18

b) 22

c) 33

d) 40

✅ Answer: c) 33

2. What is the only even prime number?

a) 3

b) 5

c) 7

d) 2

✅ Answer: d) 2

3. Which of the following pairs are co-prime?

a) (8, 12)

b) (5, 9)

c) (6, 15)

d) (12, 18)

✅ Answer: b) (5, 9)

4. What is the sum of an even and an odd number?

a) Even

b) Odd

c) Prime

d) Composite

✅ Answer: b) Odd

5. What is the HCF of two co-prime numbers?

a) 0

b) 1

c) Their product

d) Their sum

✅ Answer: b) 1

6. What is the divisibility rule for 3?

a) Check last two digits

b) Sum of digits is a multiple of 3

c) Ends in 0 or 5

d) Number is even

✅ Answer: b) Sum of digits is a multiple of 3

7. Which of the following numbers is divisible by 6?

a) 23

b) 54

c) 97

d) 81

✅ Answer: b) 54

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