Numbers are not only for counting; they are full of interesting patterns and relationships. In this chapter, we study factors, multiples, prime and composite numbers, and the divisibility rules that help us check whether one number divides another without performing actual division. These ideas help us break down big numbers into simpler parts and understand their structure.
Every number can be written as a product of smaller numbers. For example, 12 = 1 x 12 = 2 x 6 = 3 x 4. The numbers 1, 2, 3, 4, 6 and 12 are called factors of 12, and 12 is called a multiple of each of them. Factors and multiples are the two sides of the same coin. Understanding them helps us find highest common factors (HCF) and lowest common multiples (LCM), which are used in fractions, ratios and real-life problems such as arranging objects in equal rows or finding common meeting times.
We also learn about prime numbers, which have exactly two factors, and composite numbers, which have more than two factors. The Sieve of Eratosthenes helps us find all prime numbers up to a given number. Divisibility rules make our work faster, and prime factorization helps us find HCF and LCM systematically.
2. Factors and Multiples
A factor of a number is an exact divisor of that number. If a divides b completely (leaving no remainder), then a is a factor of b, and b is a multiple of a.
15 divided by 3 = 5 (exact). So 3 and 5 are factors of 15, and 15 is a multiple of 3 and of 5.
The factors of 15 are 1, 3, 5 and 15.
The first few multiples of 4 are 4, 8, 12, 16, 20 and so on.
Properties of Factors and Multiples
1 is a factor of every number.
Every number is a factor of itself.
Every number is a multiple of 1 and of itself.
Every factor of a number is less than or equal to the number.
Every multiple of a number is greater than or equal to the number.
The number of multiples of a given number is infinite.
0 is a multiple of every number.
3. Even and Odd Numbers
A number that is a multiple of 2 is called an even number. Even numbers end in 0, 2, 4, 6 or 8. Examples are 2, 4, 6, 8, 10, 100.
A number that is not a multiple of 2 is called an odd number. Odd numbers end in 1, 3, 5, 7 or 9. Examples are 1, 3, 5, 7, 9, 101.
Properties of Even and Odd Numbers
Sum of two even numbers is even. Example: 4 + 6 = 10.
Sum of two odd numbers is even. Example: 3 + 7 = 10.
Sum of an even and an odd number is odd. Example: 4 + 3 = 7.
Product of two even numbers is even.
Product of two odd numbers is odd.
Product of an even and an odd number is even.
4. Prime and Composite Numbers
A prime number has exactly two factors: 1 and itself. Examples are 2, 3, 5, 7, 11, 13, 17 and 19. The number 2 is the smallest and the only even prime number.
A composite number has more than two factors. Examples are 4, 6, 8, 9, 10 and 12.
The number 1 is neither prime nor composite because it has only one factor.
A number greater than 1 having at least one factor other than 1 and itself is composite.
Twin Primes
Pairs of prime numbers whose difference is 2 are called twin primes. Examples are (3, 5), (5, 7) and (11, 13).
Prime Triplets
Sets of three consecutive prime numbers differing by 2 are called prime triplets. The only example is (3, 5, 7).
Co-prime Numbers
Two numbers are co-prime (or relatively prime) if they have no common factor other than 1. Co-prime numbers need not be prime numbers. For example, 4 and 9 are co-prime even though both are composite.
5. Divisibility Rules
Divisibility rules help us decide quickly whether a number is divisible by another number without dividing.
Divisible by 2: The number ends in 0, 2, 4, 6 or 8. Example: 5,438.
Divisible by 3: The sum of the digits is divisible by 3. Example: 4 + 5 + 3 = 12, and 12 is divisible by 3, so 453 is divisible by 3.
Divisible by 4: The last two digits form a number divisible by 4. Example: 1,324, since 24 is divisible by 4.
Divisible by 5: The number ends in 0 or 5. Example: 7,845.
Divisible by 6: The number is divisible by both 2 and 3. Example: 486 is even and digit sum is 18, so it is divisible by 6.
Divisible by 8: The last three digits form a number divisible by 8. Example: 2,168 since 168 is divisible by 8.
Divisible by 9: The sum of the digits is divisible by 9. Example: 5,427, since 5 + 4 + 2 + 7 = 18, divisible by 9.
Divisible by 10: The number ends in 0. Example: 6,540.
Divisible by 11: The difference between the sum of digits in alternate positions is 0 or a multiple of 11. Example: 9,582, since (9 + 8) - (5 + 2) = 17 - 7 = 10, not divisible by 11.
6. Highest Common Factor (HCF)
The highest common factor (HCF) of two or more numbers is the greatest number that divides all of them exactly. It is also called the Greatest Common Divisor (GCD).
Methods to Find HCF
Method 1: List all factors of each number and pick the largest common one. For 12 and 18, factors of 12 are 1, 2, 3, 4, 6, 12 and factors of 18 are 1, 2, 3, 6, 9, 18. Common factors are 1, 2, 3, 6. The HCF is 6.
Method 2: Prime factorization. Write each number as a product of primes, then multiply the common prime factors. For 12 = 2 x 2 x 3 and 18 = 2 x 3 x 3, the common factors are 2 and 3, so HCF = 2 x 3 = 6.
Method 3: Division method. Divide the larger number by the smaller, then divide the divisor by the remainder, and continue until the remainder is zero. The last divisor is the HCF.
7. Lowest Common Multiple (LCM)
The lowest common multiple (LCM) of two or more numbers is the smallest positive number that is a common multiple of all of them.
Methods to Find LCM
Method 1: List multiples. For 6 and 8, multiples of 6 are 6, 12, 18, 24, 30 and multiples of 8 are 8, 16, 24, 32. The smallest common multiple is 24, so LCM = 24.
Method 2: Prime factorization. Take each prime factor with the highest power appearing in any number. For 6 = 2 x 3 and 8 = 2 x 2 x 2, we take 2 to power 3 and 3, giving LCM = 2 x 2 x 2 x 3 = 24.
Method 3: Common division method. Divide all numbers by common prime factors repeatedly until no common factor remains, then multiply all divisors and remaining numbers.
HCF and LCM Relationship
For two positive integers a and b: HCF(a, b) x LCM(a, b) = a x b. For example, HCF(12, 18) x LCM(12, 18) = 6 x 36 = 216 = 12 x 18.
8. Applications of HCF and LCM
HCF is used to find the largest size of an equal group. For example, the largest number of students that can be arranged equally in rows from two groups.
LCM is used to find the smallest number that satisfies certain conditions. For example, the smallest number divisible by two or more numbers, or the next common time when two events happen together.
Quick Revision Tables
Divisibility by
Rule
Example
2
Last digit is 0, 2, 4, 6 or 8
5,438
3
Sum of digits divisible by 3
453 (sum 12)
4
Last two digits divisible by 4
1,324 (24)
5
Ends in 0 or 5
7,845
6
Divisible by both 2 and 3
486
9
Sum of digits divisible by 9
5,427 (sum 18)
10
Ends in 0
6,540
Numbers 1 to 20
Type
2, 3, 5, 7, 11, 13, 17, 19
Prime
4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20
Composite
1
Neither prime nor composite
Mind Map
flowchart TD
A["Playing with Numbers"] --> B["Factors and Multiples"]
A --> C["Prime and Composite Numbers"]
A --> D["Divisibility Rules"]
A --> E["HCF"]
A --> F["LCM"]
A --> G["Even and Odd Numbers"]
C --> C1["Prime: exactly 2 factors"]
C --> C2["Composite: more than 2 factors"]
C --> C3["1: neither prime nor composite"]
E --> E1["Listing method"]
E --> E2["Prime factorization"]
E --> E3["Division method"]
F --> F1["Listing method"]
F --> F2["Prime factorization"]
F --> F3["Common division method"]
B --> B1["HCF x LCM = a x b"]
Important Diagrams (SVG)
Factor Tree
HCF of 12 and 18
Common Mistakes
Calling 1 a prime number. The number 1 is neither prime nor composite because it has only one factor.
Calling 2 a composite number. The number 2 is the smallest and the only even prime number.
Confusing the divisibility rules of 3 and 9. Both use digit sum, but 3 needs the sum divisible by 3 while 9 needs it divisible by 9.
Thinking that co-prime numbers must both be prime. Co-prime numbers only need HCF equal to 1.
Forgetting to include 1 and the number itself when listing factors.
Applying the rule of 11 incorrectly by taking the wrong order of digit positions.
Exam Tips
Memorise the divisibility rules for 2, 3, 4, 5, 6, 8, 9, 10 and 11 with examples.
For HCF and LCM questions, choose the fastest method based on the size of the numbers.
Remember that the HCF of two numbers is always less than or equal to the smaller number.
Remember that the LCM of two numbers is always greater than or equal to the larger number.
Use the relation HCF x LCM = product of numbers to verify your answers.
In word problems, decide whether the question needs HCF (greatest size) or LCM (smallest time or number).
Conclusion
Playing with Numbers reveals the hidden structure of numbers through factors, multiples, primes and divisibility. We learnt how to identify prime and composite numbers, apply divisibility rules for quick checks, and compute HCF and LCM using multiple methods. These tools are essential for working with fractions, ratios and algebra, and they help us solve many real-life problems about grouping and common times.