What does "hundred chicken skills" mean?

According to legend, during the Northern and Southern Dynasties (386-589 AD), a "child prodigy" appeared in northern China. He has quick thinking and excellent computing ability, and he has solved many problems that even adults can't answer at the moment. People from far and near like to ask him to solve math problems.

The fame of "prodigy" is getting bigger and bigger, and it has spread to the ears of the prime minister of the dynasty. One day, in order to find out whether the "child prodigy" is true or not, the Prime Minister specially called the "child prodigy" father and gave him 100 pence to bring 100 chickens the next day. It is also stipulated that 100 chickens should have roosters, hens and chicks, no more nor less, and it must be exactly 100 yuan for 100 chickens.

At that time, it took five pence to buy 1 rooster, three pence to buy 1 hen and only 1 penny to buy three chickens. How can I make a hundred chickens for a hundred dollars? The "prodigy" thought about it and told his father that all he had to do was send four cocks, 18 hens and 78 chickens.

The next day, the prime minister was surprised to see that the chickens he sent just met the demand of one hundred chickens for one hundred yuan. He thought for a moment, gave 100 pence, and sent 100 chickens tomorrow. It is also stipulated that only four cocks are allowed.

This problem has not stumped the "prodigy". He thought about it and asked his father to send eight cocks, 1 1 hen and 8 1 chicken. He also told his father that if he encountered similar problems, he just had to do what he could.

The next day, the Prime Minister was surprised to see 100 chickens. He gave 100 pence again, and asked to give 100 chickens next time.

Unexpectedly, in a short time, the "prodigy" father sent 100 chickens. Premier: 12 roosters, 4 hens and 84 chickens, just enough to meet the demand of 100 chickens for 100 yuan. ...

This "prodigy" is Zhang Qiujian. He continued to study hard and eventually grew into a famous mathematician. In his masterpiece "Zhang Qiujian Su 'an Classic", the last topic is this interesting "Hundred Chicken Problem".

"Hundred chickens problem" is an indefinite equation problem.

If the number of roosters, hens and chickens are x, y and z respectively, the equation can be obtained according to the meaning of the question:

X x y+z = 100

5x+3y x 1/3z = 100。

In addition to setting the integer parameter k, there are:

X=4k，

Y=25-7k，

Z = 75 13k。

Because the number of chickens x, y and z can only be positive numbers, the k value satisfying this set of formulas can only be 1, 2, 3. Replace k in the formula with 1, 2, 3 respectively, and the calculated answer is exactly the same as that of Zhang Qiujian.

In the era when Zhang Qiujian lived, people could not solve the indefinite equation. So, how did he come up with several answers to this question?

It turns out that Zhang Qiujian discovered a secret: four cocks are worth 20 pence and three chickens are worth 1 penny, so the total number of chickens is 7 and the money is 21; As for seven hens, the number of chickens is seven, and the money is also 2 1. If you buy seven fewer hens, you can use the money to buy four more cocks and three more chickens. In this way, a hundred chickens are still a hundred chickens, and a hundred dollars is still a hundred dollars. So as long as one answer is found, according to this law, other answers can be found immediately.

This is the famous "Hundred Chicken Technique" at home and abroad.