Comprehensive theory, key formulas, diagrams, and memory aids for Playing with Numbers.
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.
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.
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.
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.
Pairs of prime numbers whose difference is 2 are called twin primes. Examples are (3, 5), (5, 7) and (11, 13).
Sets of three consecutive prime numbers differing by 2 are called prime triplets. The only example is (3, 5, 7).
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.
Divisibility rules help us decide quickly whether a number is divisible by another number without dividing.
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).
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.
The lowest common multiple (LCM) of two or more numbers is the smallest positive number that is a common multiple of all of them.
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.
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.
| 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 |
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"]
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.