098r. Prime Factors and Divisibility Tests

Learning Intentions

• find the prime factor form of a number
• To understand how the lowest common multiple and highest common factor of two numbers can be found using their prime factor form
• use the divisibility tests for single digit factors other than 7

Pre-requisite Summary

• Know that a prime number has exactly two factors
• Be able to identify common prime numbers such as $2,3,5,11$
• Know that a composite number can be broken into prime factors
• Understand that a factor divides a number exactly
• Know the meaning of lowest common multiple and highest common factor
• Be able to list simple multiples and factors of whole numbers
• Understand that divisibility tests help decide quickly whether one number divides another exactly
• Recall divisibility tests for small numbers from place value and digit sums

Worked Examples

Worked Example 1

Write each number in prime factor form:

a) $24$
b) $45$
c) $72$

Worked Example 2

Write each number in prime factor form using index notation:

a) $36$
b) $60$

Worked Example 3

Use prime factor form to find the highest common factor of:

a) $24$ and $36$
b) $45$ and $75$

Worked Example 4

Use prime factor form to find the lowest common multiple of:

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

Worked Example 5

Use divisibility tests to decide whether each number is divisible by $2,3,4,5,6,8$ or $9$:

a) $248$
b) $315$
c) $432$

Worked Example 6

Use divisibility tests to list all single-digit factors other than $7$ that divide each number:

a) $180$
b) $256$
c) $378$

Problems

Problem 1

Write each number in prime factor form:

a) $30$
b) $54$
c) $84$

Problem 2

Write each number in prime factor form using index notation:

a) $48$
b) $90$

Problem 3

Use prime factor form to find the highest common factor of:

a) $18$ and $30$
b) $56$ and $72$

Problem 4

Use prime factor form to find the lowest common multiple of:

a) $15$ and $20$
b) $18$ and $24$

Problem 5

Use divisibility tests to decide whether each number is divisible by $2,3,4,5,6,8$ or $9$:

a) $324$
b) $550$
c) $144$

Problem 6

Use divisibility tests to list all single-digit factors other than $7$ that divide each number:

a) $144$
b) $225$
c) $648$

Exercises

Understanding and Fluency

1. Write each number in prime factor form:
   a) $18$
   b) $28$
   c) $50$
   d) $63$

2. Write each number in prime factor form using index notation:
   a) $40$
   b) $72$
   c) $100$

3. Find the highest common factor using prime factor form:
   a) $12$ and $18$
   b) $24$ and $40$
   c) $54$ and $72$

4. Find the lowest common multiple using prime factor form:
   a) $6$ and $8$
   b) $9$ and $12$
   c) $14$ and $21$

5. Use divisibility tests to decide whether each number is divisible by $2,3,4,5,6,8$ or $9$:
   a) $216$
   b) $245$
   c) $504$

6. List all single-digit factors other than $7$ that divide each number exactly:
   a) $120$
   b) $198$
   c) $360$

7. Complete each statement:
   a) A number is divisible by $3$ if …
   b) A number is divisible by $4$ if …
   c) A number is divisible by $6$ if …
   d) A number is divisible by $9$ if …

8. For each pair, find both the HCF and the LCM using prime factor form:
   a) $16$ and $24$
   b) $30$ and $45$
   c) $36$ and $48$

Reasoning

9. Explain why $2^3\times 3$ and $2\times 2\times 2\times 3$ represent the same prime factor form.

10. A student says the HCF of two numbers is found by multiplying all the prime factors in both numbers together. Explain the mistake.

11. A student says the LCM of $12$ and $18$ is $2\times 3=6$ because these are common prime factors. Explain why this is incorrect.

12. Without dividing fully, explain why $456$ is divisible by $2,3,$ and $6$ but not by $5$.

13. A student says that if a number is divisible by $8$, then it must also be divisible by $4$. Is the student correct? Explain.

Problem-solving

14. Two bells ring every $12$ minutes and every $18$ minutes. If they ring together now, after how many minutes will they next ring together?

15. Two teams have $24$ and $36$ players. What is the greatest number of equal groups that can be made so that each group has the same number of players and no players are left over?

16. A factory packs juice boxes into trays of $6$, cartons of $8$, and crates of $9$. Which of these pack sizes can be used exactly for $432$ juice boxes? Use divisibility tests.

17. A school is arranging $48$ red counters and $60$ blue counters into identical piles with no counters left over. What is the greatest number of counters in each pile if each pile must have the same composition?

18. A number is divisible by $2,3,$ and $5$, but not by $4$ or $9$. Write two possible three-digit numbers and justify your choices using divisibility tests.

19. Two traffic lights change every $15$ seconds and every $24$ seconds. After how many seconds will they change together again?

Potential Misunderstandings

• Students may stop factorising before all factors are prime
• Students may include $1$ as a prime factor
• Students may write prime factor form in a different order and think it is incorrect
• Students may confuse prime factor form with a list of all factors
• Students may think the HCF uses all prime factors from both numbers instead of only the common prime factors with the lowest powers
• Students may think the LCM uses only common prime factors instead of all prime factors with the highest powers
• Students may mix up when to choose the lowest power and when to choose the highest power
• Students may think divisibility by $6$ only requires divisibility by $2$ or $3$, instead of both
• Students may test divisibility by $4$ using the whole number instead of the last two digits
• Students may test divisibility by $8$ using the whole number instead of the last three digits
• Students may think divisibility by $5$ depends on the sum of the digits
• Students may try to apply a divisibility test for $7$ even though it is not part of this lesson