⭐ COUPON: SALE2020 ⭐ 20% OFF ON ALL WORKBOOKS*3 DAYS FREE SHIPPING ⭐ FREE RETURNS ⭐

Fail to load the data
0

Imagine a number that can be written as the sum of all digits back and forward. That number is  89 = 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9!

Isn’t that amazing? Also, 89 is a prime number with great properties as we will study next.

We will start recalling some definitions that we have widely discussed in our Prime Numbers article. If you still feel unfamiliar with these notions, we invite you to first read that article, and come back here later.

A factor of a natural number is a positive divisor of the number. A proper factor of a natural number is a factor that is different from 1 and from the number itself. For example, 98 = 2×49=14×7=1×98; thus, 1, 2, 7, 14, 49, and 98 are all factors of 98, but only 2, 7, 14, and 49 are proper factors of 98.

A natural number is called prime number if it is greater than 1, and it doesn’t have proper factors. For example, the prime immediately preceding 89 is 83, and the prime immediately following 89 is 97.

A composite number is a natural number that has proper factors. As we saw, 98 has several proper factors, thus 98 is a composite number. Contrarily, its reverse (89) is a prime number, as we will see next.

Why is 89 a prime number?

Number 89 is prime because it doesn’t have proper factors. In other words, the only factors of 89 are 1 and itself. To be sure of it, we can use the following property.

If n is a composite number, then there is a prime number less than that divides n.

Notice that , thus . Therefore, the prime numbers less than are 2, 3, 5, and 7. Moreover,

89 = (2×44) + 1
89 = (3×29) + 2
89 = (5×17) + 4
89 = (7×12) + 5

Meaning that neither of the prime numbers 2, 3, 5 nor 7 divides 89. Then, by the property above, 89 is a prime number.

On the other hand, a prime number of objects cannot be arranged into a rectangular grid with more than one column and more than one row. This is another way of verifying that 89 is a prime number:

  • For example, if we try to arrange 89 stars into a rectangular grid with nine rows, one of the columns will be incomplete.
    The same happens if we try to arrange 89 stars into a rectangular grid with any number of rows and columns greater than one.
  • The only way of arranging 89 stars into a rectangular grid, is by having a single row, or a single column. This means that 89 is a prime number!

Which class of prime number is 89?

Number 89 is the 24th prime number. It can be written, making operations with its digits, as: . Moreover, since 8 and 9 are composite, 89 is the smallest prime whose all digits are composite numbers.

Eighty-nine can be classified into several classes of primes numbers. Next, we will name three classes and see the relation of the number 89 with them.

Classes of Prime Numbers
Primoral prime It is a prime number of the form

where

are the first n prime numbers.
No
Mersenne prime It is a prime number of the form

where n is an integer.
No
Safe prime It is a prime number of the form 2p+1 where p is also a prime number. No

Let’s find out why:

  • Using the first three prime numbers 2, 3 and 5 in the primoral formula, we get
    (2 × 3 × 5) − 1 = 29 and (2 × 3 × 5) + 1 = 31. Using the first four primes 2, 3, 5, and 7, we get
    (2 × 3 × 5 × 7) − 1 = 209 and (2 × 3 × 5 × 7) + 1=211. Since the first two of the resulting numbers are less than 89, and the last two are greater than 89, then 89 doesn’t have the form of a primoral prime.
  • Notice that:


    Therefore, 89 doesn’t have the form of a Mersenne prime. However, if we use 89 as exponent, we get the 10th Mersenne prime
  • Recall that 43 and 47 are consecutive prime numbers. If we use them in the safe primes formula, we get 2(43) + 1 = 87 and 2(47 )+ 1 = 95. Since 89 is between 2(43) + 1 and 2(47) + 1, it can’t be a safe prime. However, using p=89 in the formula, we get 2(89) + 1 = 179 which is a prime number!

We invite you to read other articles on prime numbers, on our blog, to find out which other prime numbers belong to these classes.

Frequently Asked Questions

Yes, because its only factors are 1 and itself.

No, because it doesn’t have proper factors.

No, because it is between
(2 × 3 × 5) ± 1 and (2 × 3 × 5 × 7) ± 1.

No, because it is between
and .

No, because it is between
2(43) + 1 = 87 and 2(47) + 1 = 95.

What do you think about this article? Share your opinion with us

ENTER BELOW FOR ARGOPREP'S FREE WEEKLY GIVEAWAYS. EVERY WEEK!
Great! You will receive an email from US shortly. Have a great day!
FREE 100$ in books to a family!
Error! Please try again!
SUCCESS
See Related Worksheets:
1st grade
We Love Reading Time
Worksheets
 (0)
Remember how it felt when you were finally able to tell time correctly all on your own? Give your first grader...
3rd grade
Data and Drawing
Worksheets
 (0)
Drawing can be tons of fun, especially when it is related to data! In this math resource, students will have t...
1st grade
Cursive U is for Umbrella
Worksheets
 (0)
Cursive handwriting is a lost art! Students, educators, and families are being drawn back to it again, seeing ...
2nd grade
Equal Groups with Easter Eggs
Worksheets
 (0)
Students will love this "hoppy" little rabbit worksheet! They'll practice making equal groups, separating eggs...
1st grade
Math Drills - Subtracting Whole Tens
Worksheets
 (0)
Are your first graders just treading into the world of subtraction? We have the easiest learning tool for them...
Kindergarten
Right Arrow Tracing
Worksheets
 (0)
Are you ready to guide your kindergarteners and pre-school to the right path? There can’t be a better place ...

Try ArgoPrep for FREE

Learn more Try ArgoPrep for FREE

Share good content with friends and get 15% discount for 12-month subscription

Share in facebook Share in twitter

Read More

Loading content ...
Loading failed...
Exclusive Offer To Boost Your Scores!
Want 800+ Printable Math Questions?
Absolutely For Free 🥳