Lesson 1: Divisibility

Learn 1: Divisibility

Imagine that you have 13 candy bars and 15 balloons. Can you divide them equally among 3 friends? How would you check? You can check if 13 and 15 are "divisible by" 3. In other words, you can check if 13 and 15 appear in the times table of 3 or not! or find the quotient.

13 ÷ 3 = 4 R 1

Each one will take 4 candy bars and the remainder is 1.

15 ÷ 3 = 5 R 0

Each one will take 5 balloons and the remainder is 0.

So, we say that 15 is divisible by 3 because the remainder is 0 and 13 is NOT divisible by 3 because the remainder is 1.

Generally: Any number is divisible by another if the remainder of the division is zero.

Divisibility rules: Definition

Divisibility rules are a set of general rules that are often used to determine whether or not a number is absolutely divisible by another number.

Divisibility Rules

Rule of 2

All even numbers are divisible by 2. (Ends in 0, 2, 4, 6, 8)

Example: 12, 46, 780

Rule of 3

A number is divisible by 3 if the sum of its digits is divisible by 3.

Example: 93 (9+3=12, and 12 is divisible by 3).

Rule of 4

If the number formed by the last two digits is divisible by 4.

Example: 284 (84 is divisible by 4).

Rule of 5

Numbers that end in 0 or 5 are divisible by 5.

Example: 10, 45, 595

Rule of 6

Numbers divisible by both 2 and 3 are divisible by 6.

Example: 630 (even, and 6+3+0=9).

Rule of 10

If the ones digit of a number is 0, it's divisible by 10.

Example: 200, 30, 67,890

Example 1

Choose the correct answer:

Which of the following is divisible by 2?

A. 61B. 823C. 374D. 117
The number ____ is divisible by 3.

A. 335B. 235C. 1,175D. 222
The number ____ is divisible by 3 and 4?

A. 435B. 524C. 414D. 228

Solution

1. C

2. D

3. D

Example 2

Choose the correct answer:

Which of the following is divisible by 6?

A. 723B. 110C. 120D. 130
Which of the following is divisible by 10?

A. 375B. 312C. 700D. 7,002

Solution

1. C

2. C

Remarks

Each whole number is divisible by 1.
Each whole number except zero is divisible by itself.
Zero is divisible by any whole number except itself.

Learn 2: The relation between divisibility, factors and multiples

We know that 3 x 5 = 15. 3 and 5 are factors of 15 which means that:

3 x 5 = 15

• 15 is the product of the factor 3 by a whole number.
So we can say that 15 is a multiple of 3

• In the division of 15 by 3, the remainder is 0.
So we can say that 15 is divisible by 3

• 15 is the product of the factor 5 by a whole number.
So we can say that 15 is a multiple of 5

• In the division of 15 by 5, the remainder is 0.
So we can say that 15 is divisible by 5

i.e. "is a multiple of" is equivalent to "is divisible by"

Generally

If a number is a multiple of another number, then the first number is necessarily divisible by the second number.
If a number is divisible by another number, then the first number is necessarily a multiple of the second number.

Example 3

Complete the following:

If 8 x 9 = 72, then 72 is divisible by ___ or ___.
If 36 is a multiple of 9, then 36 is ___ by 9.
If 412 is divisible by 4, then 4 is called a ___ of 412.

Solution

1. 8, 9

2. divisible

3. factor