Chapter focus
Number patterns invite us to predict, test and explain. Divisibility and factors provide efficient ways to check those patterns and solve problems.
Use these original notes alongside the complete activities and exercises in your assigned Ganita Prakash edition.
1. Factors, multiples and divisibility
A factor of a whole number divides it exactly. Since 18 = 3 × 6, both 3 and 6 are factors of 18. A multiple is a product of the number and a whole-number factor: 18, 36 and 54 are multiples of 18.
- Divisible by 2: last digit is 0, 2, 4, 6 or 8.
- Divisible by 3: the sum of the digits is divisible by 3.
- Divisible by 5: last digit is 0 or 5.
- Divisible by 9: the sum of the digits is divisible by 9.
- Divisible by 10: last digit is 0.
These tests help identify factors quickly, but for an unfamiliar divisor, use division or factorisation.
2. Prime and composite numbers
A prime number is greater than 1 and has exactly two positive factors: 1 and itself. A composite number has more than two positive factors. The number 1 is neither prime nor composite. Prime factorisation expresses a composite number as a product of primes.
Example: 84 = 2 × 42 = 2 × 2 × 21 = 22 × 3 × 7. This is useful for checking divisibility and finding common factors.
3. Find and explain patterns
A pattern is more than a list of numbers. State the rule that produces each term, then test it on further terms. For 4, 7, 10, 13, … the same amount, 3, is added each time. For 2, 4, 8, 16, … each term is doubled. These patterns have different rules even if a few early terms could look similar.
To test a claim about divisibility, use examples that vary. A claim that “every number ending in 5 is divisible by 5” is true by the divisibility rule. A claim that “every odd number is prime” is false: 9 is odd but has factors 1, 3 and 9.
Worked examples
A. Test 7,326 for divisibility by 9
- Add its digits: 7 + 3 + 2 + 6 = 18.
- 18 is divisible by 9, so 7,326 is divisible by 9.
- Check: 7,326 ÷ 9 = 814 exactly.
B. List all factors of 36
Make factor pairs: 1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6. Therefore the positive factors are 1, 2, 3, 4, 6, 9, 12, 18, 36. Pairing prevents missing a factor.
Common errors to avoid
- 1 is not prime: it has only one positive factor.
- A divisibility test for 3 is not the same as the test for 9; for 9, the digit sum must itself be divisible by 9.
- Do not call a pattern a rule until you can explain how it generates every term.
- When listing factors, include both members of each factor pair.
Practice: test, then justify
1. Is 7,326 divisible by 9? Show the test.
Answer: Yes. Its digit sum is 7 + 3 + 2 + 6 = 18, which is divisible by 9.
2. List all positive factors of 18.
Answer: 1, 2, 3, 6, 9 and 18.
3. Find the prime factorisation of 180.
Answer: 180 = 18 × 10 = 22 × 32 × 5.
4. A student says 1 is prime because it is divisible by 1. Explain the mistake.
Answer: A prime must have exactly two positive factors. One has only one positive factor, so it is neither prime nor composite.
5. Continue 5, 9, 13, 17, … for three terms and state the rule.
Answer: 21, 25, 29; add 4 each time.
Chapter summary
Use factor pairs, prime factorisation and divisibility tests to reason about whole numbers. For patterns, describe the generating rule and test whether it continues to work; examples alone do not prove a general claim.