The problem of not knowing the number of things comes from China's ancient mathematical masterpiece Sun Tzu's Art of War 1600 years ago. The original title is: "Today's things are unknown, three two and three, five three and five, seven two and seven, the geometry of things?"
The meaning of this question is: there are several items, I don't know how many. If you count three, there are still two left; If you count five, there are still three left. If you count seven, there are still two left. Q: How many pieces are there in this batch?
It becomes a pure mathematical problem: there is a number, the remainder 2 is divided by 3, the remainder 3 is divided by 5 and the remainder 2 is divided by 7. Find this number.
The question is simple: 2 divided by 3, 2 divided by 7, so 2 divided by 2 1, the least common multiple of 3 and 7, 2 divided by 2 1, we will think of 23 first; 23 is divisible by 5, so 23 is an answer to this question.
This problem is simple because dividing the remainder by 3 is the same as dividing it by 7. Without this particularity, the problem would not be so simple and much more interesting.
Let's change an example; Han Xin counted the number of soldiers in a group. There were two left in a group of three, three left in a group of five and four left in a group of seven. Q: How many soldiers are there in this team?
This topic is to find a positive number, so that it can divide the remainder 2 by 3, the remainder 3 by 5 and the remainder 4 by 7. The smaller the number, the better.
If some students have never been exposed to such questions, they can also use the method of experimental analysis to increase the conditions step by step and deduce the answers.
For example, let's start with the condition that 2 is divided by 3. The number satisfying this condition is 3n+2, where n is a non-negative integer.
To make 3n+2 still meet the condition of 3 divided by 5, we can try to replace n with 1, 2, 3, … respectively. When n= 1, 3n+2=5, when 5 is divided by 5, there is no 3, which is irrelevant; When n=2, 3n+2=8, 8 divided by 5 is exactly 3. It can be seen that the number 8 satisfies both the condition of 2 divided by 3 and the condition of 3 divided by 5.
The last condition is to divide the remaining 4 by 7. 8 does not meet this condition. We want to get a number based on 8, so that it satisfies three conditions at the same time.
For this reason, people think that the new number can be equal to 8 and the sum of multiples of 3 and 5. Because 8 plus any integer multiple of 3 and 5, divided by 3 or 2, divided by 5 or 3. So we let the new number be 8+ 15m, and substitute m= 1, 2, … for the experiment. When m=3, 8+ 15m=53, and 53 divided by 7 is exactly 4, so 53 meets the requirements of the topic.
Ancient scholars in China have studied this problem for a long time. For example, Cheng Dawei, a mathematician in China in the Ming Dynasty, in his book Arithmetic Unity (1593), hinted at the solution to this problem with four very popular formulas:
The three of them lost 70 times,
Five trees, plum blossoms and a sweet branch,
Seven sons reunited for half a month.
Divide by 105.
"Full moon" means 15. The original meaning of "divide by 105" is that when the obtained number is greater than 105, it will be reduced downward by 105 and 105 to make it less than105; This is equivalent to dividing by 105 and finding the remainder.
The meanings of these four formulas are: when the divisor is 3, 5 and 7 respectively, 70 times the remainder divided by 3, 2 1 times the remainder divided by 5, 15 times the remainder divided by 7, and then add these three products. If the result of addition is greater than 105, divide by 105, and the remainder is the smallest positive integer solution that meets the requirements of the topic.
According to the method suggested by these four formulas, we can calculate the number of soldiers in Han Xin Dian's team:
70×2+2 1×3+ 15×4=263,
263=2× 105+53,
Therefore, this team has at least 53 soldiers.
In this method, we can see that 70,21and 15 are very important. After a little research, we can find that their characteristics are:
70 is a multiple of 5 and 7, 1 divided by 3;
2 1 is a multiple of 3 and 7, divided by 51;
15 is a multiple of 3 and 5, 1 divided by 7.
therefore
70×2 is a multiple of 5 and 7. Divide the remaining 2 by 3.
2 1×3 is a multiple of 3 and 7. Divide the remaining 3 by 5.
15×4 is a multiple of 3 and 5. Divide 4 by 7.
If a number is divided by A and the remainder is B, then this number plus a multiple of A is divided by A and the remainder is still B, so the results obtained by adding 70× 2,21× 3 and 15×4 can meet the requirements of "the remainder is divided by 3, the remainder by 5 and the remainder by 7" at the same time. Generally speaking,
70m+2 1n+ 15k( 1≤m < 3， 1≤n