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

Lesson 1: Multiples

In this Grade 4 lesson from The Art of Problem Solving: Prealgebra, students learn the formal definition of multiples and how to determine whether one integer is a multiple of another using quotients and remainders. The lesson covers key properties of multiples, including why the sum or difference of two multiples of a number is always a multiple of that same number. These concepts are explored through number theory problems drawn from AMC 8 competition math.

Section 1

Identify Multiples of Numbers

Property

A number is a multiple of $n$ if it is the product of a counting number and $n$ . A multiple of a number is the product of the number and a counting number.

Examples

The first five multiples of 6 are found by multiplying 6 by 1, 2, 3, 4, and 5. The multiples are 6, 12, 18, 24, and 30.

To check if 48 is a multiple of 6, we see if any counting number times 6 equals 48. Since $6 \cdot 8 = 48$ , we know 48 is a multiple of 6.

Section 2

Addition and Subtraction Properties of Multiples

Property

If $a$ and $b$ are both multiples of $c$ , then:

a + b \text{ is a multiple of } c

a - b \text{ is a multiple of } c

Algebraically: If $a = cm$ and $b = cn$ for integers $m$ and $n$ , then $a + b = c(m + n)$ and $a - b = c(m - n)$ .

Examples

Section 3

Multiples of Composite Numbers and Their Factors

Property

If $n$ is a composite number with factors $a$ and $b$ where $n = ab$ , then every multiple of $n$ is also a multiple of both $a$ and $b$ . However, not every multiple of $a$ or $b$ is necessarily a multiple of $n$ .

Examples

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Identify Multiples of Numbers

Property

A number is a multiple of $n$ if it is the product of a counting number and $n$ . A multiple of a number is the product of the number and a counting number.

Examples

The first five multiples of 6 are found by multiplying 6 by 1, 2, 3, 4, and 5. The multiples are 6, 12, 18, 24, and 30.

To check if 48 is a multiple of 6, we see if any counting number times 6 equals 48. Since $6 \cdot 8 = 48$ , we know 48 is a multiple of 6.

Section 2

Addition and Subtraction Properties of Multiples

Property

If $a$ and $b$ are both multiples of $c$ , then:

a + b \text{ is a multiple of } c

a - b \text{ is a multiple of } c

Algebraically: If $a = cm$ and $b = cn$ for integers $m$ and $n$ , then $a + b = c(m + n)$ and $a - b = c(m - n)$ .

Examples

Section 3

Multiples of Composite Numbers and Their Factors

Property

Examples

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Identify Multiples of Numbers

Property

A number is a multiple of $n$ if it is the product of a counting number and $n$ . A multiple of a number is the product of the number and a counting number.

Examples

The first five multiples of 6 are found by multiplying 6 by 1, 2, 3, 4, and 5. The multiples are 6, 12, 18, 24, and 30.

To check if 48 is a multiple of 6, we see if any counting number times 6 equals 48. Since $6 \cdot 8 = 48$ , we know 48 is a multiple of 6.

Section 2

Addition and Subtraction Properties of Multiples

Property

If $a$ and $b$ are both multiples of $c$ , then:

a + b \text{ is a multiple of } c

a - b \text{ is a multiple of } c

Algebraically: If $a = cm$ and $b = cn$ for integers $m$ and $n$ , then $a + b = c(m + n)$ and $a - b = c(m - n)$ .