Factors, Multiples, HCF and LCM
Overview
Factors, multiples, HCF and LCM form the backbone of number theory in primary and upper-primary mathematics. These concepts appear directly in UPTET Paper I and Paper II mathematics sections, and also underpin word problems involving time, work, distribution and measurement. A teacher must not only solve such problems quickly but also explain the underlying logic to young learners.
For UPTET, expect questions on identifying prime and composite numbers, applying divisibility rules, finding HCF and LCM through prime factorisation or division method, and solving application-based problems involving LCM (e.g., bells ringing together) or HCF (e.g., largest tile fitting a floor). Mastery here also strengthens your ability to simplify fractions and work with ratios—topics tested elsewhere in the syllabus.
Key Concepts
- Factor: A number that divides another number exactly (without remainder). Example: 4 is a factor of 12.
- Multiple: The product of a number and any whole number. Example: 12, 18, 24 are multiples of 6.
- Prime number: A natural number greater than 1 with exactly two factors—1 and itself. Examples: 2, 3, 5, 7, 11, 13.
- Composite number: A natural number greater than 1 with more than two factors. Examples: 4, 6, 9, 15.
- 1 is neither prime nor composite; 2 is the only even prime number.
- Co-prime (relatively prime) numbers: Two numbers whose HCF is 1. Example: 8 and 15 are co-prime.
- HCF (Highest Common Factor): The largest number that divides two or more numbers exactly. Also called GCD (Greatest Common Divisor).
- LCM (Least Common Multiple): The smallest number that is a multiple of two or more numbers.
Formulas / Key Facts
| Concept | Formula / Rule |
|---|---|
| Relationship between HCF and LCM | HCF × LCM = Product of the two numbers (for two numbers only) |
| HCF of co-primes | Always 1 |
| LCM of co-primes | Product of the two numbers |
| HCF ≤ each number; LCM ≥ each number | Always true |
| Prime factorisation for HCF | Take the lowest power of all common prime factors |
| Prime factorisation for LCM | Take the highest power of all prime factors appearing in any number |
Divisibility Rules (must memorise)
| Divisor | Rule |
|---|---|
| 2 | Last digit is 0, 2, 4, 6 or 8 |
| 3 | Sum of digits divisible by 3 |
| 4 | Last two digits form a number divisible by 4 |
| 5 | Last digit is 0 or 5 |
| 6 | Divisible by both 2 and 3 |
| 8 | Last three digits form a number divisible by 8 |
| 9 | Sum of digits divisible by 9 |
| 11 | Difference of sums of alternate digits is 0 or divisible by 11 |
Worked Examples
Example 1: Find HCF and LCM of 36 and 48 using prime factorisation.
Step 1 – Prime factorise each number
36 = 2² × 3² 48 = 2⁴ × 3¹
Step 2 – HCF (lowest powers of common primes)
Common primes: 2 and 3 HCF = 2² × 3¹ = 4 × 3 = 12
Step 3 – LCM (highest powers of all primes)
LCM = 2⁴ × 3² = 16 × 9 = 144
Verification: HCF × LCM = 12 × 144 = 1728; Product = 36 × 48 = 1728 ✓
Example 2: Three bells ring at intervals of 4, 6 and 9 minutes. If they ring together at 8:00 AM, when will they ring together next?
Solution
Find LCM of 4, 6, 9.
4 = 2² 6 = 2 × 3 9 = 3²
LCM = 2² × 3² = 4 × 9 = 36 minutes
Next simultaneous ring: 8:00 AM + 36 min = 8:36 AM
Example 3: Find the largest number that divides 120 and 90 exactly.
Solution
This asks for the HCF.
120 = 2³ × 3 × 5 90 = 2 × 3² × 5
HCF = 2¹ × 3¹ × 5¹ = 2 × 3 × 5 = 30
Example 4: Is 7429 divisible by 11?
Solution
Alternate-digit sums: (7 + 2) = 9 and (4 + 9) = 13 Difference = |9 − 13| = 4
4 is neither 0 nor divisible by 11, so 7429 is NOT divisible by 11.
Common Mistakes
| Wrong Thinking | Correct Fix |
|---|---|
| Confusing HCF and LCM—choosing the larger value for HCF. | Remember: HCF is always the smaller or equal value; LCM is always the larger or equal value. |
| Using the product formula HCF × LCM = a × b for three or more numbers. | The formula works only for two numbers. For three numbers, use prime factorisation directly. |
| Forgetting that 1 is neither prime nor composite. | 1 has only one factor (itself), so it does not meet the definition of prime (exactly two factors). |
| Ignoring a prime factor that appears in only one number when finding LCM. | LCM must include all primes from all numbers at their highest powers. |
| Applying the divisibility rule for 4 to just the last digit instead of the last two digits. | Check the number formed by the last two digits, not the last digit alone. |
Quick Reference
- HCF → largest common divisor → used for "greatest/largest" distribution problems.
- LCM → smallest common multiple → used for "together again" or "least quantity" problems.
- HCF × LCM = Product of two numbers (two numbers only).
- 2 is the only even prime; 1 is neither prime nor composite.
- Divisibility by 6 = divisible by both 2 and 3.
- Co-prime numbers share no common factor other than 1 (HCF = 1).