My IQ has been increasing year by year.

The office was completely silent.

Only the clock on the wall was ticking away.

Chen Zhuo looked at the question.

He took the pen.

That familiar, cold, metallic sensation surged from the nerves in my fingertips to the central nervous system.

He did not start writing immediately.

He broke down the problem in his mind.

The prime number p.

The index is p-2.

Divisible by.

These keywords, when combined, instantly awakened a theorem deep within his mind.

Fermat's Little Theorem.

a^(p-1)≡ 1 (mod p) (when a is not a multiple of p).

This is one of the cornerstones of number theory.

Chen Zhuo pushed up his glasses.

This question.

For junior high school students, this is indeed beyond the curriculum, and even abnormal.

Because it requires you not only to know Fermat's Little Theorem, but also to know how to flexibly apply inverses.

But in Chen Zhuo's eyes...

This is actually a very interesting question.

What is 2^(p-2)?

According to Fermat's Little Theorem, 2^(p-1)≡1(mod p).

Therefore, 2^(p-2)≡2^(-1)(mod p).

That is, the inverse of 2 modulo p.

Similarly, 3^(p-2) is the inverse of 3.

6^(p-2) is the inverse of 6.

Therefore, the problem becomes a proof:

2^(-1)+3^(-1)+6^(-1)-1≡0(mod p)。

It's too simple.

Chen Zhuo even wanted to laugh.

1/2+1/3+1/6=3/6+2/6+1/6=6/6=1

1-1 0 =

Q.E.D.

This is the beauty of mathematics.

What appears to be a complex exponential operation, under the lens of number theory, is reduced to the simplest addition and subtraction of fractions in elementary school.

Avenue to Jane.

Chen Zhuo removed the cap from his pen.

He didn't use draft paper.

He began writing directly in the blank space of the paper.

No need to draw diagrams or assume air resistance.

All you need are a few clean and concise lines of congruences.

Since p is prime, p > 3

∴(2,p)=1, (3,p)=1, (6,p)=1

By Fermat's Little Theorem:

2^(p-1)≡1(mod p)⇒2^(p-2)·2≡1(mod p)

......

Chen Zhuo wrote very quickly.

The pen scratches on the paper.

Less than two minutes.

Chen Zhuo stopped writing.

The last line.

∴ Original Expression≡1-1≡0(mod p)

QED

Chen Zhuo capped his pen and pushed the paper to Lao Zhao.

"alright."

Old Zhao remained silent, staring intently at Chen Zhuo's hands.

From the moment Chen Zhuo wrote the first congruence symbol "≡", Lao Zhao's pupils dilated.

He knew he had it in the bag.

This child not only knows how to do it, but also uses the most standard and elegant number theory language.

Instead of using a cumbersome method to expand the binomial, he directly addressed the essence of the problem.

Inverse element.

Old Zhao picked up the test paper.

Looking at those few lines of beautiful calculations.

The logical flow and the elegance of the numbers were simply perfect.

"good!"

Old Zhao slammed his hand on the table, making the teacup lid jump.

"What a brilliant Fermat's Little Theorem!"

"What a brilliant example of reverse cyclical thinking!"

Old Zhao looked at Chen Zhuo with such fervor in his eyes that it almost overwhelmed Chen Zhuo.

"I knew it."

"I knew you were born for math."

"That lousy physics teacher Zhou doesn't know anything about this kind of aesthetics."

Old Zhao stood up, took a large bunch of keys from his waistband, and searched for them for a long time.

Finally, they found a slightly rusty, brass-colored key.

He placed the key in front of Chen Zhuo.

"Take it."

Chen Zhuo looked at the key. (The teachers at this school really like giving out keys.)

"This is?"

"The key to the top floor archives."

Old Zhao's voice carried a sense of pride and boldness.

"Inside, there are all the national high school mathematics league exam papers from 85 to the present, as well as a lot of national training team materials."

"Those training camp lecture notes were all brought back by me back then, by shamelessly shoving myself, from my college classmate who was the president of the provincial mathematics society, box by box."

"Some of these materials are no longer available on the market."

"According to school regulations, that's a confidential room, and only the head of the teaching and research group can enter."

Old Zhao lowered his voice, as if he were revealing some secret.

"You can go whenever you want, and see whatever you want."

"Except for not setting it on fire, you can do whatever you want with it."

Old Zhao paused for a moment, then continued.

"besides."

"I've already registered you for next year's Hua Luogeng Gold Cup Mathematics Competition and the National Junior High School Mathematics Competition."

"From today onwards, you are also a key member of the school's math training team."

"From now on, Monday and Wednesday nights will be math training sessions, and Tuesday and Thursday afternoons are free for you to hang out with Old Zhou."

"As for the other classes..."

Old Zhao waved his hand.

"As long as your midterm and final exams are passable, I'll not only approve your leave, but anyone who has a problem with that can come find me!"

Dual privileges.

Dual-core drive.

Chen Zhuo reached out and grasped the brass key.

It's cool and a bit heavy.

"Thank you, Teacher Zhao."

Chen Zhuo stood up and bowed to Lao Zhao.

It is absolutely true.

In this day and age, it's rare to encounter two teachers who break with convention for their own sake, even competing with each other.

It was his good fortune.

Chen Zhuo walked out of the office.

He had the key to the physics lab in his left pocket and the key to the math archives in his right pocket.

They walk with a swagger.

Just as I turned the corner, I ran into someone.

Old Zhou.

Old Zhou, with his textbooks tucked under his arm, was getting ready to go to his second-year junior high class.

Seeing Chen Zhuo come out of the grade group office, and seeing Lao Zhao standing at the door looking refreshed as he saw Chen Zhuo off.

Who is Lao Zhou?

That's a shrewd person.

His squinting eyes darted around, and he immediately understood what had happened.

Old Zhou stopped and stood in front of Chen Zhuo.

"Did Old Zhao come looking for you?"

Old Zhou's voice sounded a bit sour, as if he had drunk half a jin of aged vinegar.

Chen Zhuo stopped and nodded.

"Um."

"What kind of bewitching potion did that old fox give you?"

Old Zhou glanced at Chen Zhuo's bulging right trouser pocket.

He said, "Physics is the flesh, and mathematics is the bone."

Chen Zhuo spoke the truth.

"Without bones, flesh is just rotten mud."

"Hey!"

Old Zhou was both angry and amused.

"That old man, he still has such a sharp tongue."

Old Zhou switched the book from under his arm to his other hand, looking disdainful.

"bone?"

"What kind of thing is it if it's all bones and no meat?"

"That's a skeleton! That's a specimen! That's dead!"

"Physics is alive! It has flesh and blood, it can run and jump!"

Old Zhou stretched out his finger and tapped Chen Zhuo's chest.

"Kid, don't let him fool you."

"Mathematics is a tool, a hammer, a wrench. We physics students are the ones who use tools to build machines. Have you ever seen a craftsman bowing to a hammer?"

Chen Zhuo couldn't help but laugh when he saw Lao Zhou's exasperated yet protective demeanor.

But he held back.

As a mature adult, he has a lot of experience in this kind of thing.

"teacher."

Chen Zhuo pushed up his glasses, his tone sincere.

"I think what Teacher Zhao said makes sense."

"Um?"

Old Zhou raised his eyebrows.

"If the hammer isn't good, the machine can't be made well either."

Chen Zhuo patted his right trouser pocket, where the math key was located.

"I want to sharpen the hammer first before I build your machine."

Old Zhou paused for a second.

He stared at Chen Zhuo for a few seconds.

Then he smiled.

His smile smoothed out all the wrinkles on his face.

"You little brat, you're something else."

Old Zhou chuckled and cursed, then reached out and smacked Chen Zhuo on the head.

"Trying to have your cake and eat it too, huh? Trying to please everyone, huh?"

"Good, that's ambitious."

"As long as you don't mind the fatigue, you can go back and forth between the two places."

Old Zhou waved his hand and stepped aside to make way.

"Get lost, get lost, don't delay my class."

Chen Zhuo nodded and walked away with light steps.

Old Zhou stood there, watching Chen Zhuo's back, and then glanced at Old Zhao, who was standing not far away at the office door, sneering at him.

Two old men, whose combined age was nearly a hundred, glared at each other across a corridor several meters apart.

"Old man."

"You old codger."

The two cursed almost simultaneously.

then.

They turned away, each smug and self-satisfied.

The sunlight outside the corridor was just right.

The cicadas are still chirping.