Multiples of 45

Triangular numbers are numbers that may be represented by dots in such a way that they will form an equilateral triangle – where there is equal number of dots on each side. Forty-five is one of these triangular numbers.

What makes 45 more interesting is that the sum of all single-digit numbers is 45. So, if we add numbers from 0 to 9, it will give us a sum of 45! Isn’t it amazing?

Numbers in mathematics have different meanings and significance in every one of us. But today, we will focus ourselves on understanding the number 45 and its multiples! Are you ready to join me on this journey?

Multiples of 45 are 45, 90, 135, 180 …

A multiple is the product of a number and any integer – may it be positive or negative integer. Hence, multiples of 45 can be generated by simply multiplying 45 and any number. More so, to verify if a number is indeed a multiple of 45, we need to have a quotient that is a whole number. If we are talking about multiples, remainders are not allowed!

Negative numbers can also be multiples since integers can either be negative or positive. So, numbers like -90, -45, 0, 45, 90 are examples of the multiples of 45. Furthermore, fractions cannot be used to generate multiples of 45 because they should always be a whole number.

Do you now understand the meaning of multiples? Now, let’s learn how we can find any possible multiples of 45!

How to find the multiples of 45?

Now that we know the definition of multiples of 45, it will be easier for us to solve any multiples of 45. We have two methods for generating multiples of 45 – repeated addition and multiplication method.

The first method that we have is repeated addition. This method simply means repeatedly adding 45 as many times as we want until we find the n^th multiple we are looking for.

One fact about multiples is that the first non-zero multiple of any number is itself. Hence, we can say that the first multiple of 45 is 45. So, if we are going to use the method of repeated addition in finding the 3^rd multiple of 45, we simply do this by adding 45 three times. Hence, 45 + 45 + 45 = 135.

So, using the same method, finding for the 8^th multiple of 45 will give us 45 + 45 + 45 + 45 + 45 + 45 + 45 + 45 = 360. Therefore, the 8^th multiple of 45 is 360.

While it is easy and fun doing this process, this can also take so much time, especially if we are looking for a larger n^th multiple of 45. Imagine generating the 1000^th multiple of 45 using repeated addition. It can probably take us a day to finish it.

And that’s why we have another method called multiplication method!

We have defined multiples as the product of 45 and an integer. So, any number that can be expressed as 45n where n is an integer, is a multiple of 45.

Say, if we are asked to get the 13^th multiple of 45, we can simply use the multiplication method in getting the result. Hence, $$45 \times 13 = 585$$. Therefore, the 13^th multiple of 45 is 585.

Now, can you try solving for the 107^th multiple of 45?
So, using multiplication method, $$45 \times 107 = 4,815$$.
Therefore, the 107^th multiple of 45 is 4815.

We have said that multiples can also be negative. Hence, if we are looking for the negative 5^th multiple of 45, we simply get the product of 45 and negative 5. Thus, $$45 \times -5 = –225$$. Therefore, the negative 5^th multiple of 45 is –225.

See… getting multiples of 45 is an easy task. Now, let’s take a look at this table. This table summarizes the two methods and how it is performed.

nth Multiple	Repeated Addition	Multiplication
1st multiple	45	$$45\;\times\;1\;=\;45$$
2nd multiple	45 + 45= 90	$$45\;\times\;2\;=\;90$$
3rd multiple	45 + 45+ 45= 135	$$45\;\times\;3\;=\;135$$
4th multiple	45 + 45 + 45 + 45 = 180	$$45\;\times\;4\;=\;180$$
5th multiple	45 + 45 + 45 + 45 + 45 = 225	$$45\;\times\;5\;=\;225$$

Did you know that…

…if a number is divisible by 5 and 9, then the number is also a multiple of 45?

Let’s use 180 as an example.
We know that 180 is the 4th multiple of 45. So, we need to check if it is divisible by 5 and 9.

To check if a number is divisible by 5, it should end at either 5 or 0. Since 180 ends at 0, then we know that it is divisible by 5.

Now, we need to find out if it is divisible by 9.
To say that any number is divisible by 9, the sum of its digit must be divisible by 9. The sum of the digits of 180 is given by 1 + 8 + 0 = 9. Since 9 is divisible by 9, we can conclude that 180 is also divisible by 9.

Since 180 is divisible by 5 and 9, we can confirm that this technique really works!

So, there are two things that we need to consider to say that a number is a multiple of 45:

1. Must end at either 0 or 5.
2. The sum of the digits must be divisible by 9.

Can you try to check if 978,565 is a multiple of 45?
Using the technique, we already know that it is divisible by 5 since the last digit of the given number is 5.
Now, we need to check if the sum of the digits of 978565 is divisible by 9.
So, 9 + 7 + 8 + 5 + 6 + 5 = 40. Since 40 is not divisible by 9, then 978565 is not also divisible by 9.

Therefore, 978565 is not divisible by 45.

This trick will definitely make our solving easier and faster when verifying numbers that are not multiples of 45.

List of First 30 multiples of 45

Listing all the possible multiples of 45 is pointless since there are infinitely many numbers. However, this table shows how we can generate the first 30 multiples of 45 by multiplying 45 and a positive counting number from 1 to 30.

Product of 45 and a positive counting number	Multiples of 25
$$45\;\times\;1$$	45
$$45\;\times\;2$$	90
$$45\;\times\;3$$	135
$$45\;\times\;4$$	180
$$45\;\times\;5$$	225
$$45\;\times\;6$$	270
$$45\;\times\;7$$	315
$$45\;\times\;8$$	360
$$45\;\times\;9$$	405
$$45\;\times\;10$$	450
$$45\;\times\;11$$	495
$$45\;\times\;12$$	540
$$45\;\times\;13$$	585
$$45\;\times\;14$$	630
$$45\;\times\;15$$	675
$$45\;\times\;16$$	720
$$45\;\times\;17$$	765
$$45\;\times\;18$$	810
$$45\;\times\;19$$	855
$$45\;\times\;20$$	900
$$45\;\times\;21$$	945
$$45\;\times\;22$$	990
$$45\;\times\;23$$	1035
$$45\;\times\;24$$	1080
$$45\;\times\;25$$	1125
$$45\;\times\;26$$	1170
$$45\;\times\;27$$	1215
$$45\;\times\;28$$	1260
$$45\;\times\;29$$	1305
$$45\;\times\;30$$	1350

Solving problems involving multiples of 45

Finding multiples of 45 is a lesson that we use in math, but we can also use this in solving problems related to our daily lives.

Problem #1

Marj and Kendra plan to donate their books to a non-profit organization. If they have donated seven boxes and each box contains 45 books, how many books did they contribute?

In this given problem, we need to find the total number of books Marj and Kendra donated. Since it was given that they have donated seven boxes where each box contains 45 books, we need to get the 7^th multiple of 45. So, $$45\;\times\;7\;=\;315$$.

Therefore, Marj and Kendra donated 315 books to a non-profit organization.

Problem #2

Shawn and Camilla are business partners. They sell vintage vinyl records at $45 each. In a week, they made $8,550. How many vinyl records were they able to sell?

It is stated in the problem that Shawn and Camilla were able to make $8,550 in a week by selling vinyl records at a price of $45 each. So, in order to know the total number of vinyl records they sold in a week, we need to divide the total number of earnings by the price of each vinyl record.

Hence, $$\$8550\;\div\;\$45\;=\;190$$.

Therefore, Shawn and Camilla were able to sell 190 vinyl records.

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