Analysis on the Theory of Three People's peers over 70-China's surplus

In the Ming Dynasty, there was a great mathematician named Cheng Damai who was answering the question"The number of unknown things"Question (I don't know the number of things today, the number of three is two, the number of five is three, and the number of seven is two. Ask ry ?) Use four poems to summarize the solutions to these problems:

"Three people walk 70 thin, five trees plum blossom a branch, seven son reunion is half a month, except five will know ."

This poem is the golden key to answer such questions.China Remainder TheoremOrSun Tzu's TheoremIt is a brilliant achievement of Chinese ancient mathematics.

<Classic example>

"I don't know the number of things today. There are only two of three, three of five, and three of seven. What is the geometric structure of things ?"

<Solution strategy>

Let's look for the answer from the above four poems:

70-thinDivide the remainder by 370Multiplication.
Five-tree plum blossomDivide the remainder by 521Multiplication.
Half a monthDivide the remainder by 715Left.
You will find out after, Add the above three products, minus105And the difference is obtained.

The column type is 2 × 70 + 3 × 21 + 2 × 15 = 233,233-105 × 2 = 23.

Why is 15,105, so magical?Where did 15,105, come from?

The nature of 15,105:
70 divided by more than 3 1, divided by 5, 7Therefore, 70a is divided by more than 3 a, which is also divided by 5, 7;
21 to 5 to 1, divided by 3 to 7So 21b is divided by more than 5 B, and is also divided by 3, 7;
15 divided by more than 7 1, divided by 3, 5So 15c is divided by more than 7 c and divided by 3, 5.
And 105 is the least public multiple of 3, 5, and 7..

In short, 70a + 21b + 15c is the number of c divided by 3, divided by 5, and divided by 7. If this number is large, and subtract their public multiples.

Now let's propose another solution, which is essentially the same as the above method. Please take a closer look.

Let's change the question: "I don't know the number of things today. I have two more than five, two more than seven, two more than seven, and four more than nine, ask about things ry ?"

First find the number divided by more than 9 4 ......
The number divided by more than 7 2 is: 58.
However, if 58 is divided by 5, there is no more than 2, and 58 is added with the minimum public multiple 63 of 7 and 9 until the addition is divided by 5, plus 2: 58,121,184,247 ......
Where 247 is the request.

<Finishing touch>

We actually useAddition theorem of RemainderAndSubtraction Theorem:

If the remainder of a number a divided by the divisor is A, and the remainder of a number B divided by the same divisor is B, then the remainder of A + B is a + B After Dividing A + B by the same divisor, after dividing the A-B by the same divisor, the remainder is a-B.

<Draw inferences>

1. Divide a number by more than 5 3, divided by more than 6 4, divided by more than 7 1, please find the minimum number suitable for the condition.

2. During the late Qin Dynasty, Xiao He recommended the Young Han Xin to Han Wang and worshipped Han Xin as the general. It is said that Han Xin dianbing does not require soldiers to report data, as long as soldiers queue up. Once, the team lined up with five columns with one person at the end of the line; 6 columns with five members at the end of the line; 7 columns with four members at the end of the line; and 11 lines with 10 members at the end of the line. If Han Xin has more than 2000 soldiers, how many are Han Xin soldiers?

3. What is the maximum value of a three-digit number divided by 9 + 6, 4 + 2, and 5 + 1?
9, 4, 5, two, m = 9 × 4 × 5 = 180
M1 = 20, M1 '= 5, M2 = 45, M2' = 1, M3 = 36, M3 '= 1
20*5*6 + 45*1*2 + 36*1*1 = 726
Maximum three digits: 726 + 180 = 906
 
<Integration>

4. There are dozens of holes in a circle. James starts from the hole and jumps counter-clockwise along the circle. He jumps every two holes and jumps to the B hole; if you skip every four holes, the result will only jump to the B hole. If you jump back to the hole every six holes, do you know how many holes are in the circle? (The B hole is on the right side of the hole, that is, if the number is 1 and the B hole number is exactly the number of holes)

The number of holes X, X divided by 3 and 1, divided by 5 and 1, is a multiple of 7.
3, 5, 7, two, M = 3 × 5 × 7 = 105
M1 = 35, M1 '= 2, M2 = 21, M2' = 1
35*2*1 + 21*1*1 = 91
 
General idea:Let's take a look at the holes on which James jumps every two holes. It's easy to see which holes should be ,...... That is to say, the number on the hole that James jumped to is a multiple of 3 plus 1. According to the question, James finally jumped to B, so the total number of holes is a multiple of 3 plus 1.

In the same way, if you jump to the B hole at the end of every four holes, it means that the total number of holes is a multiple of 5 and 1 is added. When you jump to a at the end of every six holes, this means that the total number of holes is a multiple of 7.

If the number of holes is reduced by 1, the number is a multiple of 3 and a multiple of 5, and thus a multiple of 15. adding 1 to the multiples of 15 is equal to the number of holes and can be divisible by 7. note that 15 is divided by 7 to 1, so 15*6 is divided by 7 to 6, and 6 of 15 is added by 1, which is exactly 7. we can also see that all the other (less than 7) multiples of 15 plus 1 cannot be divisible by 7, and 15 × 7 = 105 is already greater than, and the multiples of more than 7 do not need to be considered. therefore, the total number of holes can only be 15*6 + l = 91.
 
It's okay to use a general idea to make a small question. the superiority of Sun Tzu's theorem can be displayed only when it is used to solve a complex convolutional group.
Reprinted: http://xmuzzx.blog.hexun.com/13939890_d.html