Playing with Numbers
1. Factors and Multiples
Factor: A factor of a number divides it exactly (remainder = 0).
Multiple: A multiple of a number is obtained by multiplying it by any natural number.
Example: Factors of 12: 1, 2, 3, 4, 6, 12.
Multiples of 12: 12, 24, 36, 48, 60, ...
Key Facts:
- 1 is a factor of every number.
- Every number is a factor of itself.
- The smallest multiple of a number is the number itself.
- There are infinitely many multiples of any number.
Common Mistake: Saying 0 is a multiple. In Class 6, we consider only natural number multiples. 0 x N = 0 is technically a multiple but is excluded at this level.
2. Prime and Composite Numbers
Prime numbers: Have exactly two distinct factors: 1 and the number itself.
Composite numbers: Have more than two factors.
| Type | Definition | Examples |
|---|---|---|
| Prime | Exactly 2 factors | 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 |
| Composite | 3+ factors | 4, 6, 8, 9, 10, 12, 14, 15, 16 |
| Neither | Only 1 factor | 1 |
Note: 2 is the only even prime number. 1 is neither prime nor composite.
Exam Focus (2 marks): 'List all prime numbers between 30 and 50.'
31, 37, 41, 43, 47.
3. Divisibility Tests
| Divisible by | Test | Example |
|---|---|---|
| 2 | Last digit is even (0,2,4,6,8) | 128 ends in 8, so divisible |
| 3 | Sum of digits divisible by 3 | 471: 4+7+1=12, divisible |
| 4 | Last two digits divisible by 4 | 1324: 24 is divisible by 4 |
| 5 | Last digit is 0 or 5 | 345 ends in 5, divisible |
| 6 | Divisible by both 2 and 3 | 234: even and sum=9, divisible |
| 8 | Last three digits divisible by 8 | 3216: 216/8=27, divisible |
| 9 | Sum of digits divisible by 9 | 567: 5+6+7=18, divisible |
| 10 | Last digit is 0 | 470 ends in 0, divisible |
| 11 | Difference of alternating sums divisible by 11 | 121: (1+1)-2=0, divisible |
Worked Example: Check if 4527 is divisible by 3 and by 9.
Sum of digits = 4 + 5 + 2 + 7 = 18.
18 is divisible by 3, so 4527 is divisible by 3.
18 is divisible by 9, so 4527 is divisible by 9.
Common Mistake: Applying the test for 3 to the test for 9. The rule is the same, but the sum must be divisible by 9, not just 3.
4. HCF (Highest Common Factor)
The largest number that divides two or more numbers exactly.
Method 1: Listing factors.
Method 2: Prime factorization.
Worked Example: Find HCF of 36 and 48.
Prime factorization:
36 = 2 x 2 x 3 x 3 = 2^2 x 3^2.
48 = 2 x 2 x 2 x 2 x 3 = 2^4 x 3.
Common factors: 2^2 x 3 = 12.
HCF = 12.
Method 3: Division method.
Divide larger by smaller, then divisor by remainder, repeat until remainder = 0. The last divisor is the HCF.
5. LCM (Least Common Multiple)
The smallest positive number that is a multiple of two or more numbers.
Method: Prime factorization.
LCM = product of highest powers of ALL prime factors.
Worked Example: Find LCM of 24, 36, and 40.
24 = 2^3 x 3, 36 = 2^2 x 3^2, 40 = 2^3 x 5.
LCM = 2^3 x 3^2 x 5 = 8 x 9 x 5 = 360.
6. Relationship: HCF x LCM = Product of Numbers
For two numbers a and b: HCF(a, b) x LCM(a, b) = a x b.
Exam Focus (4 marks): 'HCF of two numbers is 12 and their product is 864. Find LCM.'
LCM = Product / HCF = 864 / 12 = 72.
7. Comparison Table: HCF vs LCM
| Feature | HCF | LCM |
|---|---|---|
| Meaning | Highest Common Factor | Least Common Multiple |
| Value | Always <= the smaller number | Always >= the larger number |
| Used in | Simplifying fractions | Adding fractions |
| Method | Common prime factors with smallest power | All prime factors with largest power |
8. Self-Test
- Write all factors of 64.
- Write first five multiples of 15.
- Check divisibility by 6: (a) 432 (b) 514 (c) 738.
- Find HCF of 54 and 90 by prime factorization.
- Find LCM of 12, 18, and 24.
- The HCF of two numbers is 8 and LCM is 120. One number is 40. Find the other.
- List all prime numbers between 1 and 30.
- Is 289 divisible by 11? Check using the divisibility rule.
9. Answers to Self-Test
- 1, 2, 4, 8, 16, 32, 64.
- 15, 30, 45, 60, 75.
- (a) 432: even and sum=9, divisible. (b) 514: even but sum=10, not divisible by 3. Not divisible by 6. (c) 738: even and sum=18, divisible.
- 54 = 2 x 3^3, 90 = 2 x 3^2 x 5. HCF = 2 x 3^2 = 18.
- 12 = 2^2 x 3, 18 = 2 x 3^2, 24 = 2^3 x 3. LCM = 2^3 x 3^2 = 72.
- Other number = (8 x 120) / 40 = 24.
- 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.
- Sum of odd places = 2 + 9 = 11. Sum of even places = 8. Difference = 3. Not divisible by 11.
