010r. Factors and Multiples

Learning Intentions

• To know what factors and multiples are
• To understand that each number has infinitely many multiples
• find factors of a number
• find multiples of a number

Pre-requisite Summary

• Understanding multiplication and division
• Knowledge of basic multiplication tables
• Ability to perform simple division and check remainders
• Understanding of even and odd numbers
• Familiarity with prime and composite numbers

Worked Examples

Worked Example 1

Identify factors:

a) Factors of $12$
b) Factors of $18$

Worked Example 2

Identify multiples:

a) First five multiples of $4$
b) First five multiples of $7$

Worked Example 3

Check if a number is a factor:

a) Is $3$ a factor of $21$?
b) Is $5$ a factor of $28$?

Worked Example 4

Check if a number is a multiple:

a) Is $24$ a multiple of $6$?
b) Is $35$ a multiple of $9$?

Problems

Problem 1

a) Find all factors of $16$
b) Find all factors of $20$

Problem 2

a) List the first five multiples of $5$
b) List the first five multiples of $8$

Problem 3

a) Is $4$ a factor of $28$?
b) Is $7$ a factor of $45$?

Problem 4

a) Is $36$ a multiple of $9$?
b) Is $50$ a multiple of $6$?

Exercises

Understanding and Fluency

1. Find factors:
   a) $10$
   b) $15$
   c) $24$

2. List multiples:
   a) First five multiples of $3$
   b) First five multiples of $6$
   c) First five multiples of $9$

3. Determine if a number is a factor:
   a) $5$ a factor of $25$
   b) $4$ a factor of $18$
   c) $7$ a factor of $42$

4. Determine if a number is a multiple:
   a) $30$ a multiple of $5$
   b) $28$ a multiple of $7$
   c) $45$ a multiple of $9$

5. Mixed practice:
   a) Factors of $36$
   b) Multiples of $4$
   c) Check if $6$ is a factor of $54$

Reasoning

6. Explain why $12$ has more factors than $11$.

7. How do you know that a number has infinitely many multiples?

8. Compare factors and multiples of $8$ and $12$.

9. Why is $1$ a factor of every number?

10. Can a prime number have more than two factors? Explain.

Problem-solving

11. Find two numbers that have exactly the same set of factors.

12. Find the first multiple of $7$ greater than $50$.

13. A gardener plants flowers in rows of $12$ and $18$. Find a number of flowers that could make complete rows for both.

14. A bus arrives every $6$ minutes, and a train every $9$ minutes. After how many minutes will they both arrive together?

15. Find all numbers less than $30$ that are multiples of $3$ and $4$.

Potential Misunderstandings

• Students may confuse factors and multiples
• Students may think a number has only a limited number of multiples
• Students may miss some factors if they stop checking early
• Students may assume all multiples are smaller than the number
• Students may forget that $1$ and the number itself are always factors
• Students may confuse prime numbers with numbers having many factors

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