Prime and composite numbers (quick recap)
A prime number has exactly two factors: 1 and itself. Examples: 2, 3, 5, 7, 11, 13. A composite number has more than two factors, like 4, 6, 60. The number 1 is neither prime nor composite. 2 is the only even prime.
What is the Fundamental Theorem of Arithmetic?
The theorem says: every composite number can be written as a product of primes, and this factorisation is unique, except for the order of the factors.
So 60 = 2 × 2 × 3 × 5 whether you start with 2 × 30, 6 × 10 or 4 × 15. You can never get a different set of primes. Think of primes as atoms: each number has its own fixed recipe.
We usually write the primes in increasing order with powers: 60 = 2² × 3 × 5.
How to find prime factors: factor tree and division
Factor tree: split the number into any two factors, keep splitting until every branch ends in a prime.
Repeated division: divide by the smallest prime that fits (2, then 3, then 5, …) until you reach 1. Example: 540 ÷ 2 = 270, ÷ 2 = 135, ÷ 3 = 45, ÷ 3 = 15, ÷ 3 = 5, ÷ 5 = 1. So 540 = 2² × 3³ × 5.
Using the theorem: can 6ⁿ end with 0?
A number ends with 0 only if it is divisible by 10 = 2 × 5, so its prime factors must include 5. But 6ⁿ = 2ⁿ × 3ⁿ has only 2 and 3. By uniqueness, no 5 can ever appear. So 6ⁿ never ends with 0 for any natural number n. The same reasoning works for 4ⁿ, 12ⁿ, etc.
HCF and LCM by prime factorisation
Write each number as a product of prime powers.
- HCF (Highest Common Factor) = product of the smallest power of each common prime.
- LCM (Lowest Common Multiple) = product of the greatest power of every prime that appears.
Example: 12 = 2² × 3 and 18 = 2 × 3². HCF = 2¹ × 3¹ = 6. LCM = 2² × 3² = 36.
HCF × LCM = product of the two numbers
For any two positive integers a and b: HCF(a, b) × LCM(a, b) = a × b. Check: 6 × 36 = 216 = 12 × 18. Why? Each prime's smaller power goes into the HCF and the larger power into the LCM, so together they use every prime exactly as a and b do.
Careful: this is not true for three numbers. For 3 numbers find HCF and LCM directly from prime factors.
Co-prime numbers
Two numbers are co-prime if their HCF is 1 (no common prime), like 8 and 15. Then LCM = a × b = 120.
Try it: a practical at home
Take 12 red and 18 blue beads (or buttons). Group each colour into prime bundles: 12 → 2, 2, 3 and 18 → 2, 3, 3. Put matching bundles side by side. The shared bundles (2 and 3) multiply to the HCF, 6. Then open the 3D above, go to the last step and try your own numbers with the sliders. Predict HCF and LCM first, then check.
Board exam corner
Real Numbers carries about 6 marks in the Number Systems unit. Common questions: find HCF and LCM by prime factorisation (2–3 marks), use HCF × LCM = a × b to find a missing number (1–2 marks), check whether a number like 6ⁿ or 4ⁿ can end with 0 (2 marks), and word problems on buses, bells or ribbons (3 marks).
Key formulas and definitions
- n = p₁^a × p₂^b × … (unique prime factorisation)
- HCF = product of smallest powers of common primes
- LCM = product of greatest powers of all primes
- HCF(a, b) × LCM(a, b) = a × b
- Co-prime: HCF = 1, LCM = a × b
Worked examples
1. Write 84 as a product of primes.
84 = 2 × 42 = 2 × 2 × 21 = 2 × 2 × 3 × 7. So 84 = 2² × 3 × 7.
2. Find the prime factorisation of 3825.
3825 ÷ 3 = 1275, ÷ 3 = 425, ÷ 5 = 85, ÷ 5 = 17 (prime). So 3825 = 3² × 5² × 17.
3. Find HCF and LCM of 24 and 90 by prime factorisation.
24 = 2³ × 3, 90 = 2 × 3² × 5. HCF = 2 × 3 = 6. LCM = 2³ × 3² × 5 = 360. Check: 6 × 360 = 2160 = 24 × 90 ✓
4. HCF of two numbers is 9 and their LCM is 360. One number is 45. Find the other.
a × b = HCF × LCM = 9 × 360 = 3240. Other number = 3240 ÷ 45 = 72.
5. Find HCF and LCM of 8, 12 and 20.
8 = 2³, 12 = 2² × 3, 20 = 2² × 5. HCF = 2² = 4. LCM = 2³ × 3 × 5 = 120. (Note 4 × 120 ≠ 8 × 12 × 20.)
6. Show that 4ⁿ can never end with the digit 0.
4ⁿ = 2²ⁿ. Ending with 0 needs a factor 5. By the Fundamental Theorem of Arithmetic, the only prime in 4ⁿ is 2, so 5 never divides it. Hence 4ⁿ never ends with 0.
7. Explain why 7 × 11 × 13 + 13 is a composite number.
7 × 11 × 13 + 13 = 13 × (7 × 11 + 1) = 13 × 78. It has factors other than 1 and itself, so it is composite.
8. Two bells ring every 20 min and 30 min. They ring together at 9:00 am. When do they next ring together?
20 = 2² × 5, 30 = 2 × 3 × 5. LCM = 2² × 3 × 5 = 60 min. They ring together again at 10:00 am.
Common mistakes
- Taking the greatest powers for HCF. HCF uses the smallest powers of common primes only.
- Forgetting a prime that appears in only one number when finding the LCM. LCM needs every prime.
- Using HCF × LCM = product for three numbers. It works only for two numbers.
- Calling 1 a prime, or stopping the factor tree at a composite leaf like 9 or 15.