Learn on PengiThe Art of Problem Solving: Prealgebra (AMC 8)Chapter 3: Number Theory

Lesson 2: Divisibility Tests

In this Grade 4 lesson from The Art of Problem Solving: Prealgebra, students learn divisibility rules for 2, 3, 4, 5, 9, and 10, including how to use digit sums and units digits to quickly determine if a number is divisible by another. The lesson also introduces the formal definition of divisibility and explains the mathematical reasoning behind each shortcut. Part of Chapter 3: Number Theory, it builds problem-solving strategies for AMC 8 preparation.

Section 1

Divisibility and Multiples

Property

A number is a multiple of $n$ if it is the product of a counting number and $n$ .
If a number $m$ is a multiple of $n$ , then $m$ is divisible by $n$ .

Examples

Section 2

Apply Divisibility Tests for 2, 3, 4, 5, 9, and 10

Property

If a number $m$ is a multiple of $n$ , then we say that $m$ is divisible by $n$ . We can use divisibility tests as shortcuts.
A number is divisible by:

2 if the last digit is 0, 2, 4, 6, or 8
3 if the sum of the digits is divisible by 3
4 if the last two digits form a number divisible by 4
5 if the last digit is 5 or 0
9 if the sum of the digits is divisible by 9
10 if the last digit is 0

Examples

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Divisibility and Multiples

Property

A number is a multiple of $n$ if it is the product of a counting number and $n$ .
If a number $m$ is a multiple of $n$ , then $m$ is divisible by $n$ .

Examples

Section 2

Apply Divisibility Tests for 2, 3, 4, 5, 9, and 10

Property

If a number $m$ is a multiple of $n$ , then we say that $m$ is divisible by $n$ . We can use divisibility tests as shortcuts.
A number is divisible by:

2 if the last digit is 0, 2, 4, 6, or 8
3 if the sum of the digits is divisible by 3
4 if the last two digits form a number divisible by 4
5 if the last digit is 5 or 0
9 if the sum of the digits is divisible by 9
10 if the last digit is 0