ENTER BELOW FOR ARGOPREP'S FREE WEEKLY GIVEAWAYS. EVERY WEEK!
FREE 100$ in books to a family!
Learning about multiples is one of the basic yet important concepts in mathematics. When we have mastered the concept of multiples, we will be able to confidently and competently tackle the more complex topics in mathematics.
This time, we will look into the multiples of another interesting number – the number 17. Aside from being a prime number, the number 17 is also the sum of the first four prime numbers, which are 2, 3, 5, 7. Moreover, it is the only prime number that is composed of consecutive prime numbers.
How awesome is that?
Are you now excited about another mathematical exploration of the multiples of 17? Gear up now because we will be digging in a whole new meaningful adventure to the world of multiples!
Multiples of 17 are 17, 34, 51, 68, 85, 102, 119, 136, 153
The multiples of a number are obtained by multiplying the number by another number. Multiples are commonly discussed in the context of integers. For example, the numbers 2, 4, 6, 8, and 10 are multiples of 2, which are obtained by multiplying 2 by the integers 1, 2, 3, 4, and 5, respectively.
Therefore, the multiples of 17 are the results we get when we multiply 17 to any integers.
Now, let us continue our mathematical exploration of how to find any multiples of 17!
Suppose n is a counting number. To get the nth multiple of 17, we must multiply 17 by n or add 17 n times. For example, if we want to determine the third multiple of 17, we must multiply 17 by 3, $$\mathbf{17}\boldsymbol\;\boldsymbol\times\boldsymbol\;\mathbf3\boldsymbol\;\boldsymbol=\boldsymbol\;\mathbf{51}$$.
It is worth noting that consecutive multiples of 17 generate a number sequence in which the difference between two consecutive numbers is 17. Thus, the multiples of 17 can also be obtained by repeated addition of 17.
So, to find the third multiple of 17 by repeated addition, we can express it as 17 + 17 + 17 = 51. Hence, the third multiple of 17 is 51.
Let’s look at the diagram below to see how easy it is to find multiples of 17 by repeated addition.
Isn’t that easy? Let us try another example by finding the sixth multiple of 17.
By multiplication, the sixth multiple of 17 can be expressed as $$\mathbf{17}\boldsymbol\;\boldsymbol\times\boldsymbol\;\mathbf6$$ and by repeated addition as17 + 17 + 17 + 17 + 17 + 17. Both methods resulted in 102. Thus, the sixth multiple of 17 is 102.
Now, look at how this table of getting the first five multiples of 17 by repeated addition and multiplication. They both have the same result, right?
1st multiple
17 × 1
17
2nd multiple
17 × 2
17 + 17
3rd multiple
17 × 3
17 + 17 + 17
4th multiple
17 × 4
17 + 17 + 17 + 17
5th multiple
17 × 5
17 + 17 + 17 + 17 + 17
Now, you can efficiently find any number that is a multiple of 17!
But wait, we must keep in mind that when a multiple of a number is divided by that number, the result is always a whole number. For example, 153 is a multiple of 17, and if we divide it by 17, the result is 9, which is a whole number. Hence, we cannot use fractions to obtain multiples. Always remember that, okay?
We have just learned that we can obtain the multiples of 17 by multiplying 17 to any integer. And since there are infinitely many integers, there are also infinitely many multiples of 17. The first 30 multiples of 17 obtained by multiplying 17 by the integers from 1 to 30 are listed in the table below
34
51
68
85
17 × 6
102
17 × 7
119
17 × 8
136
17 × 9
153
17 × 10
170
17 × 11
187
17 × 12
204
17 × 13
221
17 × 14
238
17 × 15
255
17 × 16
272
17 × 17
289
17 × 18
306
17 × 19
323
17 × 20
340
17 × 21
357
17 × 22
374
17 × 23
391
17 × 24
408
17 × 25
425
17 × 26
442
17 × 27
459
17 × 28
476
17 × 29
493
17 × 30
510
Did you know that…There is another way to test if a number with 3 or more digits is a multiple of 17? We just need to follow the simple steps below.
There is another way to test if a number with 3 or more digits is a multiple of 17? We just need to follow the simple steps below.
Let us consider the number 969. The first step is to multiply the last digit by 5 so, $$\mathbf9\boldsymbol\;\boldsymbol\times\boldsymbol\;\mathbf5\boldsymbol\;\boldsymbol=\boldsymbol\;\mathbf{45}$$. Next, we will subtract 45 from the remaining digits so, 96 – 45 = 51. Since our end result is 51 – which is the 3rd multiple of 17, then we can say that 969 is divisible by 17.
You may also try this to other numbers and see for yourself that this method works.
What a cool thing to know!
The time has come for you to put what we have learned about the multiples of 17 into practice by answering the following real-life problems!
Problem #1Daisy loves traveling in different countries and collecting souvenirs. In each of the countries she visited, she always bought five different souvenirs. These souvenirs are being displayed in the living room of her house. If she has been to 17 different countries already, how many souvenirs did she collect in total?
To solve this, we must take note that Daisy visited 17 countries and bought five souvenirs from each country. So, we must find the 5th multiple of 17 to determine her total souvenirs. To do this, we simply multiply 17 by 5. Thus, $$\mathbf{17}\boldsymbol\;\boldsymbol\times\boldsymbol\;\mathbf5\boldsymbol\;\boldsymbol=\boldsymbol\;\mathbf{85}$$.
Therefore, Daisy has a total of 85 souvenirs displayed in the living room of her house.
Did you get the same answer? Now let us try to solve another problem.
Problem #2Robert and his sister, Cathy, want to save $115 for their mother’s birthday present. They decided to have a part-time job during the summer break. Robert worked in a bakery where he earned $3 a day. Cathy worked in a coffee shop and earned $4 a day. If both of them worked for 17 days, would their total earnings together be enough to buy a birthday present for their mother?
To solve this, we must first compute the earnings of each of the siblings.
Robert’s earning is given by the 3rd multiple of 17. So, $$\mathbf{17}\boldsymbol\;\boldsymbol\times\boldsymbol\;\mathbf3\boldsymbol\;\boldsymbol=\boldsymbol\;\mathbf{51}$$. Thus, Robert earned $51.
Cathy’s earning is given by the 4th multiple of 17, which can be expressed as $$\mathbf{17}\boldsymbol\;\boldsymbol\times\boldsymbol\;\mathbf4\boldsymbol\;\boldsymbol=\boldsymbol\;\mathbf{68}$$. Thus, Cathy earned $68.
Next, we must combine the earnings of both siblings. So, 51 + 68 = 119. Hence, the total earnings of the siblings are $119, which is already greater than their target amount of $115.
Therefore, their total earnings together are enough to buy a birthday present for their mother.
Excellent! Knowing about the multiples of 17 are really useful in solving real-world problems, I’m sure you agree.
Now that you have learned the concept of multiples of 17, it is time for you to put your newfound knowledge to the test by answering the following questions.
Which of the following is a multiple of 17?
How many multiples of 17 are there in between 100 and 200?
6
What should be multiplied to 17 to get 408 as its multiple?
24
Lino needs to put a fence around her square flower garden. How many feet of fence does she need if her garden has a side length of 17 feet?
You can find the multiples of 17 by multiplying 17 with integers.
The first five multiples of 17 are 17, 34, 51, 68, and 85.
No. Since there are infinitely many numbers, you can multiply infinitely many numbers to 17 too, and thus, there are infinitely many multiples of 17 as well.
No. If we get a multiple of 17 by multiplying it by an even integer, we will get an even multiple of 17, such as 34, 68, 102, 136, and 170.
No. We can only have multiples of 7 that are also multiples of 17 if we multiply 7 by the multiples of 17, such as 119 and 238.
Shipping calculated at checkout.