At the end of Qin Dynasty, Chu and Han contended. On one occasion, Han Xin went to war with Li Feng, a general of Chu State, 1500 soldiers. After a bitter battle, the Chu army was defeated and retreated to the camp. The Han army also killed or injured four or five hundred people. So, Han Xin reorganized his military forces and returned to the base camp. When we were on a hillside, a rear unit reported that Chu cavalry were chasing us. I saw the dust flying in the distance and the sound of ShaSheng was deafening. The Han army was very tired, and then the team was in an uproar. Han Xin's military forces reached the top of the slope and saw that the enemy was less than 500. They quickly ordered the troops to meet them. He ordered three soldiers in a row and ended up with two more; Then he ordered the soldiers to line up five times, and as a result, there were three more; He ordered a platoon of seven soldiers, only to get two more. Han Xin immediately announced to the soldiers: Our army 1073 Warriors, with less than 500 enemy troops. If we are condescending, we will surely defeat the enemy in numbers. The Han army believed in its commander-in-chief, and now believed that Han Xin was a "fairy" and a "SJ". So morale is greatly boosted. At that time, flags shook, drums roared, the Han army advanced step by step, and the Chu army was in a mess. Shortly after the battle, the Chu army was defeated and fled.

First find the least common multiple of 5,9, 13 and 17 (note: because 5,9, 13 and 17 are pairwise coprime integers, the least common multiple is the product of these numbers), and then add 3 to get 9948 (person).

In Sun Tzu's calculation more than a thousand years ago, there was such an arithmetic problem:

"I don't know the number of things today. 3322, 5522, 7722. What is the geometry of things? " According to today's words: divide a number by 3 and 2, by 5 and 3, by 7 and 2, and find this number.

This kind of problem is also called "Han Xin's point soldier", which forms a kind of problem, namely the solution congruence formula in elementary number theory.

① There is a number, divided by 3 and 2, divided by 4 and 1. What is this number divided by 12?

Answer: The number divided by 3 and 2 is:

2, 5, 8, 1 1, 14, 17, 20, 23….

The remainder after dividing by 12 is:

2,5,8, 1 1,2,5,8, 1 1….

Divided by 4, the remaining 1 number is:

1, 5, 9, 13, 17, 2 1, 25, 29….

The remainder after dividing by 12 is:

1, 5, 9, 1, 5, 9,….

The remainder of a number divided by 12 is unique. Only 5 in the upper two lines is the same * * *, so the remainder of this number divided by 12 is 5.

If you change the problem of ①, you will not find the remainder divided by 12, but this number. Obviously, there are many qualified numbers, which are 5+ 12 × integers.

Integer can take 0, 1, 2, …, endless. In fact, after we find out 5, we notice that 12 is the least common multiple of 3 and 4, plus the integer multiple of 12, which are all numbers that meet the conditions. This is the division of "divide by 3 and 2, divide by 4 and 1".

② Divide a number by 3 and 2, by 5 and 3, and by 7 and 2 to find the smallest number that meets the conditions.

Solution: First list the numbers divided by 3 and 2:

2, 5, 8, 1 1, 14, 17, 20, 23, 26…

Then list the numbers divided by 5 and 3:

3, 8, 13, 18, 23, 28….

In these two columns, the first common divisor is 8.3, and the least common multiple of 5 is 15. These two conditions are combined into an integer 8+ 15. List the numbers in this series as 8, 23, 38, …, and then list the numbers divided by 7+2, 2, 9, 16.

The minimum number to meet the requirements of the topic is 23.

In fact, we have combined the three conditions in the topic into one: divide by105,23.

Then Han Xindian's soldiers are between 1000- 1500, which should be105×/kloc-0+23 =1073.

There is a similar question in China's ancient mathematical work Sun Tzu's Art of War: "There are things today, I don't know their numbers, three or three numbers, two, five or five numbers, three or seven numbers, two, ask about the geometry of things? 」

Answer: "Twenty-three"

The technique said, "Is there any geometry left in three numbers or three numbers? Answer: Five times seven times two equals 140.

Is there any reset geometry left in five or five numbers? Answer: Three times seven makes twenty-one.

What is the remaining geometry of the number 77? Answer, three times five equals fifteen.

Three times five times seven equals one hundred and five.

Then you can know that you have it, and

The sum of the remaining two of three or three numbers, the remaining three of one hundred and forty, five or five numbers, the remaining two of sixty-three and seventy-seven numbers gets 233, and then it is subtracted from 2 10 to get the final product. Where the number of three is one, the number of seventy-five is one, the number of twenty-one is one, and the number of seventy-seven is one and fifteen, that's all. 」

The author of Sunzi Suanjing and the exact date of its completion cannot be verified, but according to the verification, its completion date will not be after the Jin Dynasty. According to this research, the solution of this problem was found earlier in China than in the west, so the generalization of this problem and its solution are called China's remainder theorem.

To sum up briefly:

1. Calculate the divisible number between two numbers.

2. Calculate the divisible number of three numbers

3. Subtract the difference (sometimes a multiple) between the divisible numbers in 2 by the sum of the three divisible numbers in 1.

4 only calculate the results.

Han Xin took 1500 soldiers to fight, and four or five hundred people died, standing in a row of three people, two more; Stand in a row of five people, four more; Seven people stand in a row, plus six people. Han Xin immediately said the number: 1049.

If there is one more person, we can round him up. The number of survivors should be between 1000~ 1 100, namely:

3 times 5 times 7 times 10 minus 1= 1049 (person)