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# Multiples

The multiples of a number are the integers which contain the number n times; n \thinspace\in\thinspace\mathbb{Z}. This might sound a little confusing at first. Just read our article carefully and you will understand everything about it. If you have been asking yourself what are multiples then you are definitely right here.

## What are Multiples?

A multiple of a number contains that number a certain amount of times, and does not have any decimals. You can always divide the multiple of a number by the number and obtain an integer without any rest of the division, reminder or modulo.

Definition: If both numbers are integers, we can say that b is a multiple of a, only if b = n × a, for an integer n. If a is different from zero, this is equivalent to saying that b/a is an integer without a rest (modulo is o), and this number a is called divisor of b. For example, 88 is a multiple of 11 because 88 = 8 × 11. In this case b=88, n=8 and a=11.
Another example is 24, which is a multiple of 3 because 24 = 8 × 3. Here b=24, n=8 and a=3.

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## Multiples Properties

• First Property: If b is a multiple of a, then, a distinct from zero, is a divisor of b. This means that if a number is a multiple of another number, then the other number can also divide the first number into integers.
• Second Property: Every integer is a multiple of 1: 1 × a = a.
• Third Property: Every integer is a multiple of itself: a × 1 = a.
• Forth Property: 0 (zero) is a multiple of every number: 0 × a = 0. This property has little use though.

From the above follows that each number has at the least two multiples.

• Fifth Property: If a and b are multiples of n, then any natural number a+b, a-b as well as a × b and a / b are multiples, too.

Rules:

• All integers are multiples of themselves.
• All integers are multiples of the number 1.
• The multiples of the number 2 always end in 0, 2, 4, 6, or 8.
• When a number is a multiple of 3, then the sum of its digits is also a multiple of 3.
• The multiples of 5 end in 0 and 5.
• The multiples of 6 end in even numbers and 0, and the sum of its digits is a multiple of 3.
• In case of the multiples of 9, the sum of its digits is also a multiple of 9.

Remember that a multiple of a number contains that number n times. In other words, if you have a multiple of x, and if you divide that number by x, the result is a whole number without any rest.

## What are Submultiples?

If you have come here searching for multiples, you may in fact be looking for information about submultiples.

Definition: An integer a is a submultiple of another number b only if b is multiple of a.

For example, 35 is a multiple of 7 because 7 is repeated 5 times in 35. You can check this by multiplying seven by five which will give you thirty-five as result. You can also divide 35 by 7, to obtain 5, or divide 35 by 5 to get 7. Therefore, both 7 and five are submultiples of 35.

This test by division can be done with all integers, not just 35. In case you obtain decimals it means that the number is neither a multiple nor a submultiple.

## Submultiples Properties

From the definition of submultiples we can deduct the following properties:

• The number one is a submultiple of any other number, because the number 1 is a multiple of every number.
• A number is always a submultiple of itself, because any number is a multiple of itself.
• In case of 0 every number is a submultiple of 0, because zero is a multiple of any number.

## How to get the Multiples of a Number

If you have come here in search for how to get the submultiples of a number then you can do so in the next paragraph, but if you just want to obtain the multiples of a number then multiply that number with 0, 1, 2, 3, … until you have enough multiples.

## How to get the Submultiples of a Number

One way to obtain the submultiples of a number is by identifying its divisors. All divisors of a number are also submultiples. For your convenience you can use our calculator below to get the submultiples:

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We recommend to you the following method to check if a number is a multiple of another. Divide the number by the second number ≠ 0. If the modulo equals 0 then the number is a multiple of the other.

For example, to see if 73 is a multiple of 3 or not, we divide 72 by 3 and get 24 with a rest of 1. Therefore, we immediately know that 73 is not a multiple of 3. In the next example we test 60. To learn whether 60 is a multiple of 4 we divide 60 by 4 and get 15 without a rest (modulo = 0). So we can conclude that 60 is a multiple of 4.

To find out if a number is a multiple of another number you can also multiply the number with all natural numbers smaller than the number you are checking. If you want to know if 5 is a submultiple of 65 you have to multiply 5 by 1, 2, 3 etc. until the product equals 65 or is bigger. Equal means “is multiple”, whereas bigger means “is not multiple”. Using this method will also show you that not only 5, but also 13 is a submultiple of 65.

## How many Submultiples does a Number have?

The answer to this question varies with the particular number under consideration. If the number is prime, like, for example 17, then the number only has two submultiples, 1 and 17, but if the number is composite then it has at the least two submultiples.

## How many Multiples does a Number have?

The amount of multiples a number has is always unlimited because there is an infinite number of integers. If a and b are integers then for every multiplication b = a × n we can find a bigger multiple b = a × (n+1).

## Examples of Multiples

In this section we give you a few examples of multiples. We use the 3 and 7 which both have the multiple 21. If we divide 21 by 3 we obtain 7 and when we divide 21 by 7 we get 3.

## Examples of Submultiples

We also want to give a few examples of submultiples for which we use 49 and 777, which, despite being a big number has relatively few submultiples.

The submultiples of 49 are: 1, 7, 49
The submultiples of 777 are: 1, 3, 7, 21, 37, 111, 259, 777

## Multiples Exercises

As a small exercise try to find the multiples of 9,12, and 15:

The multiples of 9 are 0, 9, 18, 27, 36, 45 …
The multiples of 12 are 0, 12, 24, 36, 48, 60 …
The multiples of 15 are 0, 15, 30, 45, 60, 75 …

## Submultiples Exercises

Try to find the submultiples of 113: Answer: As 113 is prime you will only find two submultiples.

Use the number 255 to figure out how many submultiples it has. Answer: By counting the entries in 1, 3, 5, 15, 17, 51, 85, 255 you can see that the result is 8.

Get the submultiples of 460: Answer: Use our calculator above to verify your result.

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