012e. Divisibility Rules and Division Vocabulary

Learning Intentions

• To know the meaning of the terms divisible, divisor, dividend, quotient and remainder
• To understand that divisibility tests can be used to Check if a number is divisible by another number without performing the division
• test for divisibility by 2, 3, 4, 5, 6, 8, 9 and 10

Pre-requisite Summary

• Understanding division as sharing or grouping
• Knowledge of multiplication facts
• Understanding place value (ones, tens, hundreds, thousands)
• Ability to add digits of a number
• Understanding even and odd numbers

Worked Examples

Worked Example 1

Identify division vocabulary Use remainder :

a) Dividend

b) Divisor

c) Quotient

d) Remainder

Worked Example 2

Use divisibility tests:

a) Is divisible by ?

b) Is divisible by ?

c) Is divisible by ?

Worked Example 3

Use divisibility tests:

a) Is divisible by ?

b) Is divisible by ?

c) Is divisible by ?

Worked Example 4

Use divisibility tests:

a) Is divisible by ?

b) Is divisible by ?

c) Is divisible by and ?

Problems

Problem 1

Using remainder , identify:

a) Dividend

b) Divisor

c) Quotient

d) Remainder

Problem 2

Use divisibility tests:

a) Is divisible by ?

b) Is divisible by ?

c) Is divisible by ?

Problem 3

Use divisibility tests:

a) Is divisible by ?

b) Is divisible by ?

c) Is divisible by ?

Problem 4

Use divisibility tests:

a) Is divisible by ?

b) Is divisible by ?

c) Is divisible by and ?

Exercises

Understanding and Fluency

Exercise 1.

Identify the terms in remainder :

a) Dividend

b) Divisor

c) Quotient

d) Remainder

Exercise 2.

Test divisibility by , and :

a)

b)

c)

Exercise 3.

Test divisibility by and :

a)

b)

c)

Exercise 4.

Test divisibility by and :

a)

b)

c)

Exercise 5.

Test divisibility by :

a)

b)

c)

Exercise 6.

Determine all numbers each is divisible by from :

a)

b)

c)

Reasoning

Exercise 7.

Explain why a number divisible by must also be divisible by and .

Exercise 8.

Why must a number divisible by also be divisible by and ?

Exercise 9.

A student says is divisible by because . Explain the mistake.

Exercise 10.

Explain why divisibility tests are useful.

Problem-solving

Exercise 11.

Solve all numbers between and divisible by .

Exercise 12.

A number is divisible by and . It is less than . What could the number be?

Exercise 13.

Which numbers between and are divisible by ?

Exercise 14.

A number is divisible by , , and . It is less than . Give two examples.

Exercise 15.

Find a number greater than that is divisible by and .

Potential Misunderstandings

• Students may confuse divisor and dividend
• Students may believe divisibility tests give the quotient rather than just checking divisibility
• Students may incorrectly add digits for divisibility by or
• Students may forget that divisibility by requires divisibility by both and
• Students may think numbers ending in are only divisible by
• Students may misapply divisibility rules to the entire number instead of the last digits where required

Next: 013. Prime Numbers and Prime Factorisation