097r. Factors, Multiples and Powers

Learning Intentions

• To understand that a prime number has exactly two factors and a composite number has more than two factors
• To know the meaning of the terms square, square root, cube and cube root
• find the lowest common multiple and highest common factor of two numbers
• find the square, square root, cube and cube root of certain small whole numbers

Pre-requisite Summary

• Know that a factor divides a number exactly with no remainder
• Know that a multiple is found by multiplying a number by whole numbers
• Be able to list the factors of small whole numbers
• Be able to list the multiples of small whole numbers
• Understand that repeated multiplication can be written more efficiently
• Recall multiplication facts up to at least $12\times 12$
• Know that some numbers can be arranged into equal rows and columns
• Be familiar with using arrays or factor pairs to represent multiplication

Worked Examples

Worked Example 1

Decide whether each number is prime or composite:

a) $11$
b) $15$
c) $2$

Worked Example 2

Write the meaning of each term and evaluate where possible:

a) $6^2$
b) $\sqrt{49}$
c) $\sqrt[3]{27}$

Worked Example 3

Find the highest common factor of:

a) $12$ and $18$
b) $20$ and $30$

Worked Example 4

Find the lowest common multiple of:

a) $4$ and $6$
b) $5$ and $8$

Worked Example 5

Find each value:

a) $9^2$
b) $\sqrt{81}$
c) $4^3$

Worked Example 6

Find each value:

a) $\sqrt[3]{8}$
b) $\sqrt[3]{64}$
c) $3^3$

Problems

Problem 1

Decide whether each number is prime or composite:

a) $13$
b) $21$
c) $1$

Problem 2

Write the meaning of each term and evaluate where possible:

a) $5^2$
b) $\sqrt{36}$
c) $\sqrt[3]{125}$

Problem 3

Find the highest common factor of:

a) $16$ and $24$
b) $18$ and $27$

Problem 4

Find the lowest common multiple of:

a) $3$ and $7$
b) $6$ and $9$

Problem 5

Find each value:

a) $7^2$
b) $\sqrt{64}$
c) $2^3$

Problem 6

Find each value:

a) $\sqrt[3]{27}$
b) $\sqrt[3]{1}$
c) $5^3$

Exercises

Understanding and Fluency

1. Decide whether each number is prime or composite:
   a) $7$
   b) $9$
   c) $17$
   d) $25$

2. List all the factors of each number, then decide whether it is prime or composite:
   a) $10$
   b) $19$
   c) $28$

3. Find the highest common factor of each pair of numbers:
   a) $8$ and $12$
   b) $14$ and $21$
   c) $24$ and $36$

4. Find the lowest common multiple of each pair of numbers:
   a) $2$ and $5$
   b) $4$ and $10$
   c) $6$ and $15$

5. Find each square or square root:
   a) $4^2$
   b) ${12}^2$
   c) $\sqrt{16}$
   d) $\sqrt{100}$

6. Find each cube or cube root:
   a) $2^3$
   b) $6^3$
   c) $\sqrt[3]{8}$
   d) $\sqrt[3]{125}$

7. Complete the table:
   a) square of $3$
   b) square root of $121$
   c) cube of $4$
   d) cube root of $216$

8. Find both the HCF and the LCM of each pair:
   a) $6$ and $12$
   b) $9$ and $15$
   c) $8$ and $20$

Reasoning

9. Explain why $2$ is a prime number but $1$ is not.

10. Sofia says that $15$ is prime because it is odd. Explain why this is incorrect.

11. Which is greater, the HCF or the LCM of $4$ and $12$? Explain using factors and multiples.

12. Explain why $\sqrt{64}=8$ but $\sqrt[3]{64}=4$.

13. A student says that $5^2=10$ because $5\times 2=10$. Describe the mistake.

Problem-solving

14. Two lights flash every $6$ seconds and every $8$ seconds. After how many seconds will they flash together again?

15. Two classes are making equal groups for a sports activity. One class has $18$ students and the other has $24$ students. What is the greatest number of students that can be in each group if both classes must have the same group size?

16. A square garden has side length $9$ m. What is the area of the garden?

17. A cube-shaped box has side length $4$ cm. What is its volume?

18. Two buses leave a station every $9$ minutes and every $12$ minutes. If they leave together now, when will they next leave together?

19. A number is composite, has a square root that is a whole number, and is less than $30$. List all possible numbers.

Potential Misunderstandings

• Students may think a prime number is any odd number
• Students may think $1$ is prime because it has only one factor
• Students may forget that a prime number must have exactly two factors, not fewer and not more
• Students may confuse factors with multiples
• Students may think the highest common factor is always the larger of the two numbers
• Students may think the lowest common multiple must be one of the original numbers
• Students may confuse square with doubling
• Students may think $n^2$ means $n\times 2$ instead of $n\times n$
• Students may confuse square root with dividing by $2$
• Students may confuse cube with multiplying by $3$
• Students may think cube root means divide by $3$
• Students may mix up the notation $\sqrt{a}$ and $\sqrt[3]{a}$
• Students may not recognise that square roots and cube roots here are only whole numbers for certain values