[Must-read for examinees!] Tips for quickly identifying whether a number is prime or not | Sense of multiples

☆Is 2021 not a prime number?

Let's start with a question.

🌸Is 2021 a prime number?

The answer is ✖

When you prime factorize 2021, it becomes

2021 = 43 × 47

Remember this result.Problems related to the calendar year appear surprisingly often.

As an application example, be able to solve this kind of quadratic equation in an instant.

x^2 - 4x - 2021 = 0

☆A sense of multiples is important!

It is important to remember 2021 = 43 × 47, but it is also important to have the sense to instantly see that 2021 cannot be divided by simple numbers like 3.The skill to identify what a number is a multiple of (= finding divisors) is extremely important in arithmetic and mathematics.

Below, I will summarize the sense of multiples you should acquire.

🌸Multiples of 2 (even numbers)
This is easy.You only need to look at the ones digit.If the ones digit is an even number, the number itself is also an even number. Since the ones digit of 2021 is an odd number, 2021 is also an odd number.

🌸If the sum of the digits is divisible by 3, the number itself is also divisible by 3!
This is important knowledge.

Let's find the sum of the digits of 2021. 2+0+2+1 = 5

Since 5 is not a multiple of 3, 2021 is not a multiple of 3.

By the way, for 2019, 2+0+1+9 = 12

12 is a multiple of 3. Therefore, 2019 is also a multiple of 3, meaning it is divisible by 3!

Let's try dividing it. 2019 ÷ 3 = 673

See? It was divisible, wasn't it?

By the way, 673 is a prime number, so 2019 = 3 × 673

This concludes the prime factorization.

Note that this trick is only practically useful for multiples of 3 and 9.

Please be aware that this does not hold true for every number.

For those who are good at math or are taking exams for competitive schools, it is a great study exercise to try proving why this method works for multiples of 3 using variables.

🌸 Multiples of 4
Just look at the last two digits. If the last two digits are divisible by 4, the number itself is also divisible by 4.

For example, in 2020, 20 is divisible by 4, so 2020 is also divisible by 4.

🌸 Multiples of 5
This is also simple. If the last digit is 0 or 5, the number is a multiple of 5.

🌸 Multiples of 6
For this, we utilize the fact that 6 = 2 × 3.

In other words, if a number is both a multiple of 2 and a multiple of 3, it is a multiple of 6.

Let's consider 2028, for example. Since the last digit is an even number, it is a multiple of 2.

And since 2+0+2+8 = 12, it is also a multiple of 3.

This means it is also a multiple of 6.

Let's check. 2028 ÷ 6 = 338. It divided perfectly!

🌸 Multiples of 8
Check if the last three digits are divisible by 8.

For example, in 2200, the last three digits 200 are divisible by 8, so 2200 is also divisible by 8.

2200 ÷ 8 = 275.

🌸 Multiples of 9
You can use the same method as for multiples of 3.

If we consider the sum of the digits of 4167, we get 4+1+6+7 = 18, and since 18 is a multiple of 9, 4167 is divisible by 9.

4167 ÷ 9 = 463.

🍃Bonus
Knowing this much is sufficient, but I will add just a little more.

🌸For multiples of 10, you just need to check if the ones digit is 0.

🌸Multiples of 7, 11, and 13 are hard to notice unless you know them, so it is good to look them up once. It is good to go up to about 200, like 7, 14, 21...

For example, 147=7×7×3, 209=11×19, 91=7×13, 143=11×13

being able to think of these instantly will give you a significant advantage in arithmetic and mathematics.

By the way,

🌸1001=7×11×13

knowing this might also be useful.

☆How to find prime numbers up to 100

Assuming you have mastered the above, it will deepen your understanding if you actually find the prime numbers up to 100 once before the entrance exam. You can do it in an instant.

First, please prepare a table up to 100 like the one below.

(You can make it quickly using Excel, but printing out this image is also fine)

Now, please try doing it yourself.

🌸Step 1: 1 is not a prime number, so cross it out with an X. (← Be careful, as many people mistakenly think 1 is a prime number!)

🌸Step 2: 2, 3, 5, and 7 are prime numbers, so circle them and leave them as they are.

🌸Step 3: Multiples of 2 other than 2 (even numbers) are not prime numbers. Cross them all out with an X. (It is faster if you draw a line vertically)

🌸Step 4: Multiples of 5 other than 5 (even numbers) are not prime numbers. Cross them all out with an X. Multiples of 10 are already crossed out in Step 3, so in practice, you just need to draw a line vertically for 15 and below.

I think it is easy up to this point, but from here on, you need to proceed carefully so as not to make mistakes.

🌸Step 5: Multiples of 3 other than 3 are also not prime numbers. Cross them all out with an X.

🌸Step 6: Finally, please cross out all multiples of 7 except for 7 itself.That said, most of them are already crossed out, so there aren't many left to actually cross out in this step (there should only have been two of them...). Be careful not to make any mistakes...

That's all for now.

At this point, everything that hasn't been crossed out is a prime number.

Just to be sure, let's check the answers.

🐣"2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97"

If you have these 25 numbers, you are correct!

It is important to actually try it out with your own hands, so please be sure to do it yourself.

☆ How do you check if a number is prime or not?

Now, here is a slightly more advanced problem.

🌸 In the process above, don't you wonder, "Why is it okay to stop after crossing out multiples of 7? Don't I need to cross out multiples of 11?"

Actually, there is a good reason for this.Understanding this reason properly becomes very important when checking whether a number is prime or not. Please make sure you understand it.

Checking "whether a number is prime" is nothing more than checking "whether it has any divisors other than 1 and itself." This is clear, right?

And, if we assume there is a divisor other than 1 and the number itself, one of those divisors must be smaller than the "square root of that number." That is the case.

This is because if you multiply the square root by itself, you get that number, so if there were no divisors smaller than the square root, multiplying them would exceed the number itself.

This time, we looked for prime numbers up to 100.

The square root of 100 (the positive one, root 100) is 10.

Therefore, it is sufficient to cross out only the multiples of prime numbers smaller than 10, which is the square root of 100, and you don't need to worry about multiples of 11 is the reason.

...Was that difficult? But it is an important concept, so please make sure you understand it.

In general, to check if a number is prime, you only need to check if it is divisible by any prime number less than its square root..

🌸 Approximately what is the square root of 2021?

Having this kind of intuition is also important. Since 44 squared is 1936 and 45 squared is 2025, it is a little less than 45. (You can easily find the square root using the SQRT function in Excel. A more accurate value is approximately 44.955.)

Therefore, if you try dividing by primes less than 44, "2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43", and none of them divide it, then 2021 is a prime number. When you try it, it is divisible by 43.

Even if a number other than 2021 appears on the exam, you can determine whether it is a prime number using this method!✨

☆ A little chat

The method I introduced this time is a simplified version of what is called the "Sieve of Eratosthenes".

Eratosthenes was an ancient Greek scholar, and besides the "Sieve of Eratosthenes," he is also known for calculating the size of the Earth quite accurately for that time (around 240 BC). You will learn more details in high school "Earth Science." Look forward to it! 😊

As a method for prime number determination, the "Sieve of Sundaram," which is an improvement on the Sieve of Eratosthenes, is also known. If you are interested, please Google it (after your entrance exams are over).

Also, prime numbers are used in creating ciphers called RSA encryption, and they have become quite important when considering security in this IT era. For this reason, it is a subject often taken up in entrance exams and the like.

To explain it very roughly, for example, it is easy to multiply two prime numbers 89 and 97 to get 8633. But looking at 8633 and realizing that it is the product of 89 and 97 (prime factorization) is extremely difficult, isn't it?Even with 2021, it was hard to realize it was the product of 43 and 47 unless you already knew, right?

By utilizing this fact, you can create ciphers that are easy to create but difficult to solve. Math is amazing!✨

☆ Summary: I want to live like a prime number

🍎 When 2021 is prime factorized, it becomes 2021 = 43 × 47. Let's remember this result!

🍎 If the sum of the digits of a number is divisible by 3, that number is also divisible by 3! In addition, let's master the main ways to identify multiples!

🍎 1 is not a prime number! Be careful not to make a mistake!!

🍎 Generally, to check if a number is prime, you only need to check if it is divisible by any prime number less than its square root!

🍎 It is easy to multiply two prime numbers 89 and 97 to get 8633, but it is extremely difficult to realize that 8633 is the product of 89 and 97 (prime factorization).Utilizing this property, prime numbers are used in the creation of ciphers.

This is sudden, but I love prime numbers. Being divisible only by 1 and itself is a cool way to live, isn't it? lol💦😝

Dividing by 1 is actually the same as doing nothing, so it means it is only divisible by itself.

Am I the only one who senses something like "I will not be told what to do by others, I will live based on my own beliefs" from prime numbers?

Without yielding to peer pressure, let's live like prime numbers—"without knowing our place"!

Don't read the room, read a book! See you later!💕

🌸🍃The authors of this article, Study Partner, were Coper & Aya🐣

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