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)
• Solve 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 Use factors:

a) HCF of and

• Factors of :
• Factors of :
• Common factors: → HCF =

Worked Example 2

Find LCM using multiples:

a) LCM of and

• Multiples of :
• Multiples of :
• Common multiples: → LCM =

Worked Example 3

HCF using prime factorization:

a) and

• Common prime factors: → HCF =

Worked Example 4

LCM using prime factorization:

a) and

• Take highest powers: → LCM =

Problems

Problem 1

a) Find HCF of and

b) Find HCF of and

Problem 2

a) Find LCM of and

b) Find LCM of and

Problem 3

a) HCF of and using prime factorization

b) HCF of and using prime factorization

Problem 4

a) LCM of and using prime factorization

b) LCM of and using prime factorization

Exercises

Understanding and Fluency

Exercise 1.

Find HCF using factors:

a) and

b) and

c) and

Exercise 2.

Find LCM using multiples:

a) and

b) and

c) and

Exercise 3.

Find HCF using prime factorization:

a) and

b) and

c) and

Exercise 4.

Find LCM using prime factorization:

a) and

b) and

c) and

Reasoning

Exercise 5.

Explain why HCF is always a factor of both numbers.

Exercise 6.

Explain why LCM is always a multiple of both numbers.

Exercise 7.

Compare the HCF and LCM of and . What pattern do you notice?

Exercise 8.

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

Exercise 9.

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

Exercise 10.

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

Problem-solving

Exercise 11.

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

Exercise 12.

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

Exercise 13.

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

Exercise 14.

Find two numbers with HCF and LCM .

Exercise 15.

A printing press produces copies every and 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