You multiples of an integer are a set whose elements are taken after themultiplication of this fixed number for all whole numbers. Once we fix an integer and multiply it by all the integers, we will form a subset of those numbers, because every element of this set of multiples is also an element of the set of numbers whole.
Read too: Properties of multiplication that facilitate mental calculation
Multiples of an integer
consider two whole numbers known, p and q. The number p will be a multiple of q if and only if there is an integer m, so that:
p = q · m
Thus, the set of multiples of the number p can be obtained by multiplying p by all the whole numbers, the results of this operation will be the multiples of p.
Example
The first 15 multiples of 3.
To determine this set, simply multiply the first 15 whole numbers by 3.
3 · 1 = 3
3 · 2 = 6
3 · 3 = 9
3 · 4 = 12
3 · 5 = 15
3 · 6 = 18
3 · 7 = 21
3 · 8 = 24
3 · 9 = 27
3· 10 = 30
3 · 11 = 33
3 · 12 = 36
3 · 13 = 39
3 · 14 = 42
3 · 15 = 45
So the first 15 multiples of 3 are:
M(3) = {3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45}
See that we only found the first 15 multiples of 3. Since we have to multiply the 3 by all the whole numbers, the set of multiples is infinite.
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How to check if a number is a multiple of another?
To check if one number is a multiple of another, we must find an integer in such a way that the multiplication between them equals the first number. Look:
Examples
The) To check if the number 110 is a multiple of 11, we must look for an integer, which, multiplied by 11, results in 110. If it exists, the number 110 will be a multiple of 11, otherwise it will not be.
110 = 11 · 10
B) Is the number 143 a multiple of 12?
The number 143 is not a multiple of 12 since:
132 = 12 · 11
144 = 12 · 12
Note that there is no integer between 11 and 12, so there is no number that, when multiplied by 12, results in 143, so the number 143 is not a multiple of 12.
See too: Multiples and dividers: what are they and how to find them?
solved exercises
question 1 – Write all natural numbers less than 100 and multiples of 15.
Resolution
We know that the multiples of 15 are the results of multiplying the number 15 by all the integers. Since the exercise asks you to write the natural numbers less than 100 and multiples of 15, we demos multiply the number 15 by all numbers greater than zero until you find the largest multiple before 100, like this:
15 · 1 = 15
15 · 2 = 30
15 · 3 = 45
15 · 4 = 60
15 · 5 = 75
15 · 6 = 90
15 · 7 = 105
Therefore, natural numbers less than 100 and multiples of 15 are:
{15, 30, 45, 60, 75, 90}
question 2 – What is the largest multiple of 5 between 100 and 1001?
Resolution
To determine the largest multiple of 5 between 100 and 1001, simply determine the first multiple of 5 backwards.
1001 is not a multiple of 5, as there is no integer that, multiplied by 5, results in 1001.
1000 is a multiple of 5, since 1000 = 5 · 200.
Therefore, the largest multiple of 5, between 100 and 1001, is the number 1000.
by Robson Luiz
Maths teacher
