Required Mathematical Intelligence

Chapter 104 The most magical number in the world: 142857
Seemingly ordinary numbers, why is he the most amazing?Let's multiply it from 1 to 6 to see:

142857 1 × = 142857
142857 2 × = 285714
142857 3 × = 428571
142857 4 × = 571428
142857 5 × = 714285
142857 6 × = 857142
The same number, just switched positions, appeared repeatedly.So, kids, do you know how much it is multiplied by 7?

[Answer: We will surprisingly find that 999999,

And

142 857=999
14 28 57 = 99
Finally, we multiply 142857 with 142857
The answer is: 20408122449 What is the number of the first five digits and the last five digits?
20408 122449=142857
About the miraculous answer

"142857"

It was found in the Egyptian pyramid. It is a set of magical numbers. It proves that there are 7 days in a week. Once it accumulates itself, its 6 numbers will take turns in sequence. On the 7th day, they will have a holiday. 999999 goes to substitute, the number is getting bigger and bigger, every more than one week reincarnation, each number needs to be cloned once, you don't need a computer, as long as you know its clone method, you can know the answer to continue to accumulate, it has even more amazing Places are waiting for you to discover!Perhaps, it is the code of the universe...

142857×1=142857 (original number)

142857×2=285714 (rotation)
142857×3=428571 (rotation)
142857×4=571428 (rotation)
142857×5=714285 (rotation)
142857×6=857142 (rotation)
142857×7=999999 (9 will be replaced by [-] during holidays)

142857×8=1142856（7分身，即分为头一个数字1与尾数6，数列内少了7）

142857×9=1285713 (4 avatar)

142857×10=1428570 (1 avatar)

142857×11=1571427 (8 avatar)

142857×12=1714284 (5 avatar)

142857×13=1857141 (2 avatar)

142857×14=1999998 (9 also needs to be bigger)
keep counting...

The singular sum of the above numbers is "9".There may be a big secret hidden.

以上面的金字塔神秘数字举例：1 4 2 8 5 7=27=2 7=9；您瞧瞧，它们的单数和竟然都是“9”。依此类推，上面各个神秘数，它们的单数和都是“9”；怪也不怪！（它的双数和27还是3的三次方）无数巧合中必有概率，无数吻合中必有规律。何谓规律？大自然规定的纪律！科学就是总结事实，从中找出规律。

Take a number at will, for example, 48965, sum the numbers of this number, the result is 4 8 9 6 5=32, and then sum the results, get 3 2=5. I call this summation method To find the mode sum of a number.

All numbers have the following rules:
（1）众数和为9的数字与任意数相乘，其结果的众数和都为9.例如306的众数和为9，而306×22=6732，数字6732的众数和也为9（6 7 3 2=18，1 8=9）。

（2）众数和为1的数字与任意数相乘，其结果的众数与被乘数的众数和相等。例如13的众数和为4，325的众数和为1，而325×13=4225，数字4225的众数和也为4（4 2 2 5=13，1 3=4）。

（3）总结得出一个普遍的规律，如果A·B=C，则众数和为A的数字与众数和为B的数字相乘，其结果的众数和亦与C的众数和相等。例如3×4=12.取一个众数和为3的数字，如201，再取一个众数和为4的数字，如112，两数相乘，结果为201×112=22512，22512的众数和为3（2 2 5 1 2=12，1 2=3），可见3×4=12，数字12的众数和亦为3.
（4）另外，数字相加亦遵守此规律。例如3 4=7.求数字201和112的和，结果为313，求313的众数和，得数字7（3 1 3=7），刚好3与4相加的结果亦为7.
It is strange that the ancient Chinese knew this mathematical law long ago.Let's take a look at the digital maps of "Hetu" and "Luoshu".The following is the digital map of "Lua Shu".

Everyone in the world knows that the "Luoshu" number diagram is famous because it is the earliest magic square diagram in the world. It is characterized by adding any set of numbers, and the result is 15. Its practical number mode and To analyze this graph, we will find that any random combination of numbers multiplied with each other will result in a mode sum of 9. For example, a random combination of numbers in the first row is 924, and a random combination of numbers in the second row is The random combination number is 159, and the result is 146916 when the two are multiplied together. Find the mode sum and get 1 4 6 9 1 6=27, 2 7=9. It can be seen that the mode sum of the result is 9.
The number map of "Hetu" is not as famous as the number map of "Luoshu". This is because people have not discovered its mathematical laws, but they can find its wonders by analyzing it with the law of the sum of mode numbers.

In the "Hetu" digital diagram, if any group of numbers is multiplied with each other, the mode sum of the result is 6. For example, 27165×38495=1045716675, find the mode sum of the result, 1 4 5 7 1 6 6 7 5 =42, 4 2=6, it can be seen that the mode sum of the result is 6.
It can be seen that the number map of "Hetu" cannot be arranged randomly, otherwise, the mode and impossibility of the result are both 6. From the above two number maps, we can see that the ancients attached great importance to the number 6 and the number 9. , the Taiji diagram is composed of the number 6 and the number 9.

The left part of the Taiji diagram is the number 6, and the right part of the Taiji diagram is the number 9.
"Tai Chi Tu", "He Tu" and "Luo Shu" imply the importance of the number 6 and the number 9 through various means, among which "He Tu" and "Luo Shu" were compiled on the premise of familiarity with the modes and laws of numbers made. ]
(End of this chapter)