Prime Factors — Solutions

1. Identify Prime Numbers

   1. 7: Prime ▶ View Solution
   2. 9: Composite — factor: 3 ▶ View Solution
   3. 11: Prime ▶ View Solution
   4. 15: Composite — factor: 3 or 5 ▶ View Solution
   5. 17: Prime ▶ View Solution
   6. 21: Composite — factor: 3 or 7 ▶ View Solution
   7. 23: Prime ▶ View Solution
   8. 27: Composite — factor: 3 ▶ View Solution

2. Complete Factor Trees

   1. 12: 2² × 3 ▶ View Solution
   2. 20: 2² × 5 ▶ View Solution
   3. 36: 2² × 3² ▶ View Solution
   4. 48: 2⁴ × 3 ▶ View Solution
   5. 60: 2² × 3 × 5 ▶ View Solution
   6. 72: 2³ × 3² ▶ View Solution
   7. 84: 2² × 3 × 7 ▶ View Solution
   8. 100: 2² × 5² ▶ View Solution

3. Write Prime Factorisation in Index Form

   1. 18: 2 × 3² ▶ View Solution
   2. 24: 2³ × 3 ▶ View Solution
   3. 32: 2⁵ ▶ View Solution
   4. 40: 2³ × 5 ▶ View Solution
   5. 54: 2 × 3³ ▶ View Solution
   6. 75: 3 × 5² ▶ View Solution
   7. 90: 2 × 3² × 5 ▶ View Solution
   8. 120: 2³ × 3 × 5 ▶ View Solution

4. Highest Common Factor (HCF)

   1. HCF of 12 and 18: 6 ▶ View Solution
   2. HCF of 24 and 36: 12 ▶ View Solution
   3. HCF of 30 and 45: 15 ▶ View Solution
   4. HCF of 48 and 72: 24 ▶ View Solution
   5. Explain how to find HCF: Take the lowest power of each shared prime factor, then multiply ▶ View Solution

5. Lowest Common Multiple (LCM)

   1. LCM of 4 and 6: 12 ▶ View Solution
   2. LCM of 8 and 12: 24 ▶ View Solution
   3. LCM of 5 and 7: 35 ▶ View Solution
   4. LCM of 6 and 10: 30 ▶ View Solution
   5. Explain how to find LCM: Take the highest power of each prime factor that appears in either number, then multiply ▶ View Solution

6. Problem Solving with Prime Factors

   1. Value of 2³ × 3 × 7: 168 ▶ View Solution
   2. Value of 2² × 5²: 100 ▶ View Solution
   3. Bells ring together (LCM of 12 and 18 min): 9:36 am ▶ View Solution
   4. Why is 1 not prime?: Unique factorisation would break down — every number would have infinitely many factorisations ▶ View Solution
   5. Is 2⁴ × 3² a perfect square?: Yes — equals 12² = 144 ▶ View Solution

7. Larger Numbers — Factor Trees and Factorisations

   1. 126: 2 × 3² × 7 ▶ View Solution
   2. 144: 2⁴ × 3² ▶ View Solution
   3. 180: 2² × 3² × 5 ▶ View Solution
   4. 252: 2² × 3² × 7 ▶ View Solution
   5. 360: 2³ × 3² × 5 ▶ View Solution
   6. 500: 2² × 5³ ▶ View Solution
   7. 630: 2 × 3² × 5 × 7 ▶ View Solution
   8. 1 000: 2³ × 5³ ▶ View Solution

8. HCF and LCM of Three Numbers

   1. HCF of 12, 18 and 24: 6 ▶ View Solution
   2. LCM of 4, 6 and 9: 36 ▶ View Solution
   3. HCF of 30, 45 and 60: 15 ▶ View Solution
   4. LCM of 6, 8 and 12: 24 ▶ View Solution
   5. LCM of 12 and 18: 36 ▶ View Solution

9. Perfect Squares via Prime Factorisation

   1. All exponents (2, 2, 2) are even → perfect square. Value: 4 × 9 × 25 = 900 = 30² ▶ View Solution
   2. 72 = 2³ × 3². Exponent of 2 is odd (3) → multiply by 2. Result: 2⁴ × 3² = 144 = 12² ▶ View Solution
   3. All exponents (6, 4) are even → perfect square. Value: 2⁶ × 3⁴ = 64 × 81 = 5184 = 72² ▶ View Solution
   4. (i) 2⁴ × 3²: all even → perfect square (= 144).   (ii) 2³ × 5²: exponent 3 is odd → not a perfect square.   (iii) 2² × 7⁴: all even → perfect square (= 4 × 2401 = 9604) ▶ View Solution
   5. Common primes: 2, 3, 5. Lower powers: 2³, 3¹, 5¹. HCF = 8 × 3 × 5 = 120 ▶ View Solution

10. Real-World Applications of HCF and LCM

    1. Muffins to exactly fill both 18 and 24-muffin trays: 72 muffins ▶ View Solution
    2. Largest square tile for 12 cm × 18 cm floor: 6 cm ▶ View Solution
    3. When buses meet again (every 20, 30 and 45 min): 180 minutes (3 hours) ▶ View Solution
    4. Longest equal fence sections for 48 m and 36 m sides: 12 m ▶ View Solution