Multiples of a Number

Multiples of a number are the products you get when that number is multiplied by any integer, including zero.

In other words, a multiple of a number is any result obtained by multiplying it by an integer.

If you have a number $ a $ and multiply it by an integer $ n $, you get a multiple of $ a $: $$ \text{Multiple of } a = a \cdot n $$ where $ n $ is a natural number (0, 1, 2, 3, …).

For instance, the multiples of 5 are:

$$ 5 \cdot 0 = 0 \\ 5 \cdot 1 = 5 \\ 5 \cdot 2 = 10 \\ 5 \cdot 3 = 15 \\ 5 \cdot 4 = 20 \\ \dots $$

This shows that the multiples of any natural number are endless, as you can keep multiplying $ a $ by larger and larger numbers.

For example, the multiples of 3 are 0, 3, 6, 9, 12, 15, … and the sequence goes on indefinitely.

Key Properties of Multiples
Understanding the Difference Between Multiples and Divisors

Key Properties of Multiples

Multiples have some essential properties that make them useful in mathematics.

Every number is a multiple of itself
For any number $ a $, one of its multiples is always $ a \cdot 1 = a $. This means that every number is a multiple of itself.
For example, 7 is a multiple of 7.
0 is a multiple of every number
Multiplying any number by zero results in zero, so $ 0 $ is a multiple of every number. This is because zero acts as the absorbing element in multiplication.
For example, $ 1 \cdot 0 = 0 $, $ 2 \cdot 0 = 0 $, $ 3 \cdot 0 = 0 $ ...
Multiples of 1
The multiples of 1 include all natural numbers and integers, as multiplying 1 by any natural number (or integer) simply gives that number. This happens because 1 is the identity element in multiplication.

For example, $ 1 \cdot 1 = 1 $, $ 2 \cdot 1 = 2 $, $ 3 \cdot 1 = 3 $ ...

Understanding the Difference Between Multiples and Divisors

A number $ b $ is a divisor of $ a $ if $ a $ can be divided by $ b $ with no remainder, while a multiple of $ a $ is a number you get by multiplying $ a $ by a natural number.

So, if $ b $ divides $ a $ with no remainder, then $ a $ is a multiple of $ b $.

To clarify:

Divisor: A number $ b $ is a divisor of $ a $ if dividing $ a $ by $ b $ gives an integer quotient without a remainder. Formally, we can say $ b $ divides $ a $ if there’s an integer $ q $ such that: $$ a = q \cdot b $$
Multiple: $ a $ is a multiple of $ b $ if it can be written as $ a = b \cdot n $, where $ n $ is a natural number. This means $ a $ is part of the sequence of multiples of $ b $, generated by multiplying $ b $ by different natural values.

Let’s look at a specific example. Suppose $ a = 12 $ and $ b = 3 $. Dividing $ 12 $ by $ 3 $ gives a quotient of $ 4 $ with no remainder $ 12 \div 3 = 4$. Because there’s no remainder, we can say $ 3 $ is a divisor of $ 12 $. Likewise, $ 12 $ is a multiple of $ 3 $, since $ 12 = 3 \cdot 4 $.

This relationship also works in reverse: if $ a $ is a multiple of $ b $, then $ b $ is a divisor of $ a $.

For instance, $ 12 $ is a multiple of $ 3 $, so $ 3 $ is a divisor of $ 12 $.

Example of divisor-multiple relationship

This complementarity makes the divisor-multiple concept especially useful for various mathematical applications, such as number factorization, finding common multiples, and understanding prime numbers.