Arithmetic That Is Neither Calculation Power Nor Knowledge - Multiples and Divisors (Part 2) +

In the previous article, we started with the story of a commercial that said it was impossible to divide 700 million among 5 people, and developed it into an example of thinking along the lines of the characteristics of multiples of 9, which seemed somewhat disconnected.

This time, I will introduce an example question from the Itsuki Shobo junior high school mock test in mathematics, which was given in 2021. It is for third-year junior high school students.

Hearing that, you might have had a rejection reaction, thinking, 'It's too hard, I don't get it!', but this is a type of problem where 'thinking' is fun, so please give it a try.

First, let me make a disclaimer.
It's not about the answer, but since this is something learned in junior high school, it is assumed that you already know the term 'prime factorization'.

It was about expressing numbers as a product of prime numbers, like:
12 = 2 x 2 x 3
45 = 3 x 3 x 5
If you 'prime factorize' the numerical examples given at the beginning of the problem statement, they become:
21 = 3 x 7
77 = 7 x 11

Now, please prime factorize 2021!

Don't try to escape by saying, 'Wait, isn't 2021 a prime number!?' at a glance. You can definitely factorize it.

Can it be divided by 3? No.
Can it be divided by 7? No.
Then, is it 11? No.
Could it be 13? ... 'Okay, stop! Put your pencils down'.

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Of course, the introduction in the problem statement has a meaning.

Both I and my wife, who is a former math teacher (is this the first time I've revealed that?), did not know the term 'cousin primes' and it was our first time seeing it.
Of course, for junior high school students (for the average student), this is their first encounter with it during this test, so everyone is answering under the same conditions of 'not knowing/not having learned in textbooks' about the term itself and its meaning.

In other words,

Even if you are not familiar with it / even if you don't know it because you studied and mastered it,

Without being flustered by the way this is presented, you have no choice but to use your reading comprehension and move forward 👟

◆2021 is the product of cousin primes
◆The difference between those two prime factors is 4

Since that is the case, 2021 = a x (a+4), so what is a?

It is just a problem like that. But actually, there is no need to grasp it mathematically as shown above.

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　Now then. Why don't we think about it logically like this?

・First, I want to make a rough estimate
・"4" is not a very large number, so...
・Let's square "something" that I can calculate mentally

"50x50=2500" --- Too big 💦
"40x40=1600" --- Too far below 2021... But, it's definitely in the 40s!

・Even numbers (other than 2) are not prime numbers, so we can exclude them, leaving only odd numbers

{41, ,43, ,45, ,47, ,49}

・The two prime numbers that have a difference of 4 from each other are...
It must be 43 and 47 🤔 Let's try doing the long multiplication...
43 x 47 = 2021 Bingo 🎯

Therefore,
the answer to ① Prime factorize 2021: 43x47

And for ② how many divisors does 2021 have? part.
You don't need to be able to do ①.
Just by accepting the explanation of "cousin primes" as it is it is solved immediately ☝

{1, a, b, 2021}

It is said that there are 4 of them. Just "translate the question"!

　From that perspective,
"..., find the answer" the sentence (phrasing) is bothersome~ 🥴
I don't have the feeling of "finding" it. It is self-evident ☝

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If you, as an adult, have the impression that "I'm bad at arithmetic/math! It's impossible!," it is simply because the context until now has been a bit off. If you have come closer to the impression that, as shown here, there are questions that can be solved (or rather, must be solved) simply by using reading comprehension and logical reasoning—in other words, 【comprehensive language skills are being tested】—then I am happy that my contribution was worthwhile.

You need to appropriately understand a concept that is suddenly presented and then expand your thinking to "In that case, it means this, right?" It is not academic ability, butthe power to think. It is not about "learning and knowing." Knowledge itself is not very useful.

《A trivial bonus I found while researching》
・A pair of prime numbers with a difference of 4 is called "cousin primes." → That seems to be derived from the existence of something called "twin primes" (difference of 2)... And furthermore...
・A pair of prime numbers with a difference of 6 is called "sexy primes"🥰
　

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