011. Highest Common Factor (HCF) and Lowest Common Multiple (LCM)

Learning Intentions

• To know the meaning of the terms highest common factor (HCF) and lowest common multiple (LCM)
• find the highest common factor of two numbers
• find the lowest common multiple of two numbers

Pre-requisite Summary

• Understanding factors and multiples of numbers
• Knowledge of prime numbers and prime factorization
• Ability to list all factors and multiples of small numbers
• Understanding multiplication and division

Worked Examples

Worked Example 1

Find HCF using factors:
a) HCF of $12$ and $18$

• Factors of $12$: $1,2,3,4,6,12$
• Factors of $18$: $1,2,3,6,9,18$
• Common factors: $1,2,3,6$ → HCF = $6$

Worked Example 2

Find LCM using multiples:
a) LCM of $4$ and $6$

• Multiples of $4$: $4,8,12,16,20,\dots$
• Multiples of $6$: $6,12,18,24,\dots$
• Common multiples: $12,24,\dots$ → LCM = $12$

Worked Example 3

HCF using prime factorization:
a) $24$ and $36$

• $24=2^3\times 3$
• $36=2^2\times 3^2$
• Common prime factors: $2^2\times 3=12$ → HCF = $12$

Worked Example 4

LCM using prime factorization:
a) $8$ and $12$

• $8=2^3$
• $12=2^2\times 3$
• Take highest powers: $2^3\times 3=24$ → LCM = $24$

Problems

Problem 1

a) Find HCF of $20$ and $30$
b) Find HCF of $42$ and $56$

Problem 2

a) Find LCM of $5$ and $10$
b) Find LCM of $6$ and $8$

Problem 3

a) HCF of $36$ and $54$ using prime factorization
b) HCF of $45$ and $60$ using prime factorization

Problem 4

a) LCM of $9$ and $12$ using prime factorization
b) LCM of $14$ and $20$ using prime factorization

Exercises

Understanding and Fluency

1. Find HCF using factors:
   a) $16$ and $24$
   b) $18$ and $30$
   c) $21$ and $35$

2. Find LCM using multiples:
   a) $3$ and $5$
   b) $4$ and $10$
   c) $6$ and $8$

3. Find HCF using prime factorization:
   a) $48$ and $60$
   b) $36$ and $84$
   c) $54$ and $72$

4. Find LCM using prime factorization:
   a) $6$ and $15$
   b) $8$ and $14$
   c) $12$ and $18$

Reasoning

5. Explain why HCF is always a factor of both numbers.

6. Explain why LCM is always a multiple of both numbers.

7. Compare the HCF and LCM of $8$ and $12$. What pattern do you notice?

8. Why does prime factorization make finding HCF and LCM easier for larger numbers?

9. If two numbers are co-prime, what is their HCF? Explain.

10. If one number is a multiple of the other, what is the LCM? Explain.

Problem-solving

11. Two ropes measure $36$ m and $60$ m. What is the longest length that can measure both exactly?

12. Two buses arrive every $12$ minutes and $18$ minutes. After how many minutes will they arrive together?

13. A gardener wants to plant flowers in rows of $8$ and $12$. What is the minimum number of flowers to make complete rows for both?

14. Find two numbers with HCF $6$ and LCM $72$.

15. A printing press produces copies every $24$ and $36$ minutes. When will both presses finish a batch together?

Potential Misunderstandings

• Students may confuse factors and multiples when finding HCF or LCM
• Students may forget to take the highest powers for LCM in prime factorization
• Students may forget that HCF uses the lowest powers of common primes
• Students may list multiples incorrectly or stop too early
• Students may assume HCF is always smaller than both numbers without checking
• Students may confuse co-prime numbers with numbers having no common multiples

Next: 012e. Divisibility Rules and Division Vocabulary