Tests of Divisibility

P6 Mathematics - Term I, Topic 4

Lesson 1 of 4

Learning Objectives:

• Identify numbers divisible by 2, 3, and 5
• Apply divisibility rules to test large numbers
• Classify numbers as even or odd

The Problem

If I have 4,536 books to share equally among 3 classrooms, will each get the same number with none left over?

Do we need to divide 4,536 by 3?

NO! There's a faster way!

We can use divisibility rules!

What Does "Divisible" Mean?

A number is divisible by another number if:

• It can be divided exactly
• There is no remainder

Examples:

• 12 ÷ 3 = 4 (no remainder) → 12 IS divisible by 3
• 14 ÷ 3 = 4 remainder 2 → 14 is NOT divisible by 3

Even and Odd Numbers Review

Even Numbers: 0, 2, 4, 6, 8, 10, 12...

• Can be divided by 2 exactly

Odd Numbers: 1, 3, 5, 7, 9, 11, 13...

• Cannot be divided by 2 exactly

Quick Tip: Just look at the LAST DIGIT!

Divisibility Rule for 2

A number is divisible by 2 if its LAST DIGIT is:

0, 2, 4, 6, or 8

(These are the even digits!)

Testing Divisibility by 2

Just check the last digit!

Practice: Divisibility by 2

Pause and Practice

Is each number divisible by 2?

1. 4,532 → Last digit is ____ → ____
2. 7,845 → Last digit is ____ → ____
3. 98,760 → Last digit is ____ → ____

Practice: Answers

1. 4,532 → Last digit is 2 → YES
2. 7,845 → Last digit is 5 → NO
3. 98,760 → Last digit is 0 → YES

Divisibility Rule for 5

A number is divisible by 5 if its LAST DIGIT is:

0 or 5

(Think: 5, 10, 15, 20, 25, 30... all end in 0 or 5!)

Testing Divisibility by 5

Just check the last digit!

Practice: Divisibility by 5

Pause and Practice

Is each number divisible by 5?

1. 6,785 → Last digit is ____ → ____
2. 9,042 → Last digit is ____ → ____
3. 100,000 → Last digit is ____ → ____

Practice: Answers

1. 6,785 → Last digit is 5 → YES
2. 9,042 → Last digit is 2 → NO
3. 100,000 → Last digit is 0 → YES

Divisibility Rule for 3

A number is divisible by 3 if the SUM OF ITS DIGITS is divisible by 3

This one is different - we need to ADD all the digits!

Testing Divisibility by 3: Example 1

Is 123 divisible by 3?

Step 1: Add all the digits
1 + 2 + 3 = 6

Step 2: Is 6 divisible by 3?
6 ÷ 3 = 2 → YES!

Answer: 123 IS divisible by 3 ✓

Testing Divisibility by 3: Example 2

Is 4,536 divisible by 3?

Step 1: Add all the digits
4 + 5 + 3 + 6 = 18

Step 2: Is 18 divisible by 3?
18 ÷ 3 = 6 → YES!

Answer: 4,536 IS divisible by 3 ✓

Testing Divisibility by 3: Example 3

Is 2,345 divisible by 3?

Step 1: Add all the digits
2 + 3 + 4 + 5 = 14

Step 2: Is 14 divisible by 3?
14 ÷ 3 = 4 remainder 2 → NO!

Answer: 2,345 is NOT divisible by 3 ✗

Practice: Divisibility by 3

Pause and Practice

Is each number divisible by 3?

1. 111 → 1 + 1 + 1 = ____ → ____
2. 456 → 4 + 5 + 6 = ____ → ____
3. 1,234 → 1 + 2 + 3 + 4 = ____ → ____

Practice: Answers

1. 111 → 1 + 1 + 1 = 3 → YES (3 ÷ 3 = 1)
2. 456 → 4 + 5 + 6 = 15 → YES (15 ÷ 3 = 5)
3. 1,234 → 1 + 2 + 3 + 4 = 10 → NO (10 ÷ 3 = 3 r 1)

Summary of Divisibility Rules

Testing for Multiple Divisors

Is 30 divisible by 2, 3, AND 5?

By 2? Last digit is 0 (even) → YES
By 5? Last digit is 0 → YES
By 3? 3 + 0 = 3, and 3 ÷ 3 = 1 → YES

30 is divisible by 2, 3, AND 5!

Practice: Multiple Tests

Pause and Practice

Test 450 for divisibility by 2, 3, and 5

By 2? Last digit is ____ → ____
By 5? Last digit is ____ → ____
By 3? 4 + 5 + 0 = ____ → ____

Practice: Answer

450

By 2? Last digit is 0 → YES
By 5? Last digit is 0 → YES
By 3? 4 + 5 + 0 = 9 (9 ÷ 3 = 3) → YES

450 is divisible by all three!

Word Problem 1

Farmer's Eggs

A farmer has 1,236 eggs. Can he pack them equally into trays of 3?

Test for divisibility by 3:
1 + 2 + 3 + 6 = 12
12 ÷ 3 = 4 ✓

YES! The eggs can be packed equally.

Word Problem 2

School Groups

There are 4,580 learners. Can they be divided equally into groups of 5?

Test for divisibility by 5:
Last digit is 0 ✓

YES! They can form equal groups of 5.

Quick Assessment ✓

1. What do you check to test divisibility by 2?

2. What do you check to test divisibility by 5?

3. What do you check to test divisibility by 3?

4. Is 2,463 divisible by 3?

Quick Assessment: Answers

1. The last digit (must be 0, 2, 4, 6, or 8)

2. The last digit (must be 0 or 5)

3. The sum of all digits (must be divisible by 3)

4. 2 + 4 + 6 + 3 = 15, and 15 ÷ 3 = 5 → YES

Remember the Rules!

These tests work for ANY size number!

Homework

Test each number for divisibility by 2, 3, and 5:

1. 234
2. 1,005
3. 7,890
4. 12,345
5. 111,111

Create a table with your answers!

Well Done!

You can now test divisibility quickly!

Next lesson: Square Numbers

Keep practicing!