Question: 11 a Write each of these numbers as a product of prime numbers. i 15 ii 15² iii 28 iv 28² v 36 vi 36² b What do you notice about your answers to i and ii, iii and iv, v and vi? c If 96 = 2⁵ x 3, show how to find the prime factors of 96². Will your method work for all numbers? 12 40 = 2 x 2 x 2 x 5 and 28 = 2 x 2 x 7 Use these facts to find a the HCF of 40 and 28 b the LCM of 40 and 28. 13 450 = 2 x 3 x 3 x 5 x 5 and 60 = 2 x 2 x 3 x 5 Use these facts to find a the HCF of 450 and 60 b the LCM of 450 and 60. 14 180 = 2² x 3² x 5 and 54 = 2 x 3³ Use these facts to find a the HCF of 180 and 54 b the LCM of 180 and 54. 15 a Write 45 as a product of prime numbers. b Write 75 as a product of prime numbers. c Find the LCM of 45 and 75. d Find the HCF of 45 and 75. 16 a Draw factor trees to find the LCM of 90 and 140. b Compare your answer with a partner's. Did you draw the same factor trees? Have you both got the same answer? 17 a Write 396 as a product of prime numbers. b Write 168 as a product of prime numbers. c Find the HCF of 396 and 168. d Find the LCM of 396 and 168. 18 a Find the HCF of 34 and 58. b Find the LCM of 34 and 58. 19 Show that the HCF of 63 and 110 is 1. 20 37 and 47 are prime numbers. a What is the HCF of 37 and 47? b What is the LCM of 37 and 47? c Write a rule for finding the HCF and LCM of two prime numbers. d Compare your answer to part c with a partner's answer. Check your rules by finding the HCF and LCM of 39 and 83.

1. **Problem:** Write each number as a product of prime numbers. 2. **Prime factorization:** - i) $15 = 3 \times 5$ - ii) $15^2 = (3 \times 5)^2 = 3^2 \times 5^2$ - iii) $28 = 2^2 \times 7$ - iv) $28^2 = (2^2 \times 7)^2 = 2^4 \times 7^2$ - v) $36 = 2^2 \times 3^2$ - vi) $36^2 = (2^2 \times 3^2)^2 = 2^4 \times 3^4$ 3. **Observation:** - Squaring a number squares each prime factor's power. 4. **Prime factors of $96^2$ given $96 = 2^5 \times 3$:** - $96^2 = (2^5 \times 3)^2 = 2^{10} \times 3^2$ - This method works for all numbers because $(a \times b)^2 = a^2 \times b^2$. 5. **HCF and LCM of 40 and 28:** - $40 = 2^3 \times 5$ - $28 = 2^2 \times 7$ - HCF uses minimum powers: $2^2$ - LCM uses maximum powers: $2^3 \times 5 \times 7$ - So, HCF = $4$, LCM = $280$ 6. **HCF and LCM of 450 and 60:** - $450 = 2 \times 3^2 \times 5^2$ - $60 = 2^2 \times 3 \times 5$ - HCF = $2^{\cancel{1}} \times 3^{\min(2,1)} \times 5^{\min(2,1)} = 2 \times 3 \times 5 = 30$ - LCM = $2^{\max(1,2)} \times 3^{\max(2,1)} \times 5^{\max(2,1)} = 2^2 \times 3^2 \times 5^2 = 900$ 7. **HCF and LCM of 180 and 54:** - $180 = 2^2 \times 3^2 \times 5$ - $54 = 2 \times 3^3$ - HCF = $2^{\min(2,1)} \times 3^{\min(2,3)} = 2 \times 3^2 = 18$ - LCM = $2^{\max(2,1)} \times 3^{\max(2,3)} \times 5 = 2^2 \times 3^3 \times 5 = 540$ 8. **Write 45 and 75 as prime products:** - $45 = 3^2 \times 5$ - $75 = 3 \times 5^2$ 9. **LCM and HCF of 45 and 75:** - HCF = $3^{\min(2,1)} \times 5^{\min(1,2)} = 3 \times 5 = 15$ - LCM = $3^{\max(2,1)} \times 5^{\max(1,2)} = 3^2 \times 5^2 = 225$ 10. **Factor trees for 90 and 140 (conceptual):** - $90 = 2 \times 3^2 \times 5$ - $140 = 2^2 \times 5 \times 7$ - LCM = $2^2 \times 3^2 \times 5 \times 7 = 1260$ 11. **Write 396 and 168 as prime products:** - $396 = 2^2 \times 3^2 \times 11$ - $168 = 2^3 \times 3 \times 7$ 12. **HCF and LCM of 396 and 168:** - HCF = $2^{\min(2,3)} \times 3^{\min(2,1)} = 2^2 \times 3 = 12$ - LCM = $2^{\max(2,3)} \times 3^{\max(2,1)} \times 7 \times 11 = 2^3 \times 3^2 \times 7 \times 11 = 2772$ 13. **HCF and LCM of 34 and 58:** - $34 = 2 \times 17$ - $58 = 2 \times 29$ - HCF = $2$ - LCM = $2 \times 17 \times 29 = 986$ 14. **HCF of 63 and 110:** - $63 = 3^2 \times 7$ - $110 = 2 \times 5 \times 11$ - No common prime factors, so HCF = $1$ 15. **HCF and LCM of 37 and 47 (both prime):** - HCF = $1$ - LCM = $37 \times 47 = 1739$ 16. **Rule for two primes:** - HCF is always $1$ - LCM is the product of the two primes 17. **Check rule with 39 and 83:** - $39 = 3 \times 13$ - $83$ is prime - HCF = $1$ - LCM = $39 \times 83 = 3237$