# Divisibility Tests

Achievement Objectives
NA4-1: Use a range of multiplicative strategies when operating on whole numbers.
Specific Learning Outcomes

Use divisibility rules for 2, 3, 4, 5, 6, 8, 9.

Description of Mathematics

Number Framework Stage 7

Required Resource Materials
Divisibility Tests (Material Master 8-24)
Activity

Important work involving finding factors of numbers such as Lowest Common Multiples (LCMs), Highest Common Factors (HCFs), and the addition or subtraction of fractions is simplified if students have some tests to spot simple factors like 2, 3, 4, 5, 6, and 10.

#### Using Number Properties

Problem: “Prakesh believes that 2 is a factor of a number only when the last digit is even. Explore this claim. Is Prakesh correct?” (Answer: Yes.)

Problem: “Prakesh writes down the multiples of 5 in order: 5, 10, 15, 20, 25, .... He spots a way of telling whether a number has 5 as a factor or not. What does he spot?”

(Answer: All multiples of 5 end in 0 or 5.)

Problem: “To find whether 4 is a factor of 34 984, Prakesh has an inspiration. He splits the number into 34 900 and 84. He is certain 34 900 has 4 as a factor. Why?”

(Answer: 34 900 = 349 x 100 and 100 ÷ 4 = 25. So 34 900 has a factor 4.)

“84 has 4 as a factor because 84 ÷ 4 has no remainder. So explain why 34 984 has 4 as a factor.”

Examples: Which of these numbers are multiples of 4? 345 638,  232,  12 002,
1 295 904, 180 008 ...

Problem: “Explore this claim: 9 is a factor only if the digit sum is divisible by 9.”

(Answer: It is correct. This is hard to show. An example:
2 745 = 2 x 1 000 + 7 x 100 + 4 x 10 + 5
= 2 x 999 + 7 x 99 + 4 x 9 + (2 + 7 + 4 + 5)
= 9 x (2 x 111 + 7 x11 + 4) + (2 + 7 + 4 + 5)

So 9 is a factor of 9 x (2 x 111 + 7 x 11 + 4). So 9 is a factor of 2 745 only if 9 is a factor of (2 + 7 + 4 + 5), which it is.)

Similarly 3 is a factor of a number only when the digit sum is divisible by 3.

Examples: Worksheet (Material Master 8–24).

#### Understanding Number Properties:

Explain how you would test whether 36 is afactor of a number.