Seemingly ordinary numbers, why is he the most magical? Let's take a look at it by multiplying it from 1 to 6

142857X1=142857

142857X2=285714

142857X3=428571

142857X4=571428

142857X5=714285

142857X6=857142

The same number, just reversed, appears over and over again.

So how much is multiplying it by 7? We will be surprised to find that it is 999999

And 142 + 857 = 999

14+28+57=99

Finally, we use 142857 multiplication with 142857

The answer is: What is the number of the top five digits + the last five digits of 20408122449?

20408+122449=142857

===Split line===

The magical answer to it

“142857”

It is found in the Egyptian pyramids, it is a set of magic numbers, it proves that there are 7 days in a week, it accumulates itself once, it is rotated by its 6 numbers, in order to rotate once, on the 7th day, they will be on vacation, by the 999999 to substitute class, the number is getting bigger and bigger, every more than a week of reincarnation, each number needs to be separated once, you don't need a computer, as long as you know its doppelganger method, you can know the answer to continue to accumulate, it has more magical places waiting for you to discover! Perhaps, it is the password 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 substitutes on holidays)

142857×8=1142856 (7 doppelgangers, i.e. divided into the first number 1 and the mantissa 6, with 7 missing from the sequence)

142857×9=1285713 (4 doppelgangers)

142857×10=1428570 (1 doppelganger)

142857×11=1571427 (8 clones)

142857×12=1714284 (5 clones)

142857×13=1857141 (2 clones)

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

Keep counting......

The sum of the singular numbers of the above numbers is "9". It is possible that a big secret is hidden.

Take the above pyramid mystery number as an example: 1+4+2+8+5+7=27=2+7=9; You see, their singular sum is actually "9". And so on, the sum of the singular numbers of the mystical numbers above is "9"; Blame it or not! (its even number and 27 or 3 to the third power) there must be probability in countless coincidences, and there must be rules in countless coincidences. What is a pattern? Discipline prescribed by nature! Science is all about summarizing facts and finding patterns from them.

Take any number, for example, take 48965, sum the numbers of this number, the result is 4+8+9+6+5=32, and then sum the results to get 3+2=5. I call this method of summing the sum of the modes of a number.

All numbers have the following pattern:

[1] A number whose sum is 9 is multiplied by any number, and the sum of the mode is 9. For example, the sum of the modes of 306 is 9, and 306*22=6732, the sum of the modes of the number 6732 is also 9 (6+7+3+2=18, 1+8=9).

[2] A number whose sum is 1 is multiplied by any number, and the result is that the mode is equal to the sum of the mode of the multiplied number. For example, the sum of the modes of 13 is 4,325, and the sum of the modes of 325*13=4225 is also 4 (4+2+2+5=13, 1+3=4).

[3] It is concluded that if A*B = C, then the sum of the mode is multiplied by the number of the mode and the sum of the b, and the sum of the mode is also equal to the sum of the mode of C. For example, 3*4=12. Take a number whose mode is 3, such as 201, and then take a number whose mode is 4, such as 112, multiply the two numbers, and the result is 201*112=22512, and the sum of the modes of 22512 is 3 (2+2+5+1+2=12, 1+2=3), so that 3*4=12, and the sum of the modes of the number 12 is also 3.

[4] In addition, the addition of numbers also follows this rule. For example, 3+4=7. Finding the sum of the numbers 201 and 112 gives us 313, and finding the sum of the modes of 313 gives us the number 7 (3+1+3=7), and the result of adding exactly 3 and 4 is also 7.

Curiously, the ancient Chinese knew about this mathematical law for a long time. Let's take a look at the "River Chart" and "Vena" number charts. The following is a digital diagram of "Vena".

492

357

816(Vena)

As everyone in the world knows, the reason why the "Vena" number chart is famous is because it is the world's earliest magic square diagram, which is characterized by the addition of any group of numbers, and the result is 15. Analyze this diagram according to the law of the sum of the modes of the numbers, and you will find that the random combination of any group of numbers multiplies each other, and the sum of the modes of the result is 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 159, multiplying the two, the result is 146916, and finding the sum of its modes gives 1+4+6+9+1+6=27, 2+7=9, and it can be seen that the sum of the modes of the results is 9.

The magical "missing 8 numbers".

12345679, there is a lack of 8 in this number, which we call "missing 8 numbers".

At first, I thought that this "missing 8 numbers" was only the wonder of "all colors". Who knew that after some research on the information, it was found that it still has many surprising features.

One, all the same

Former Philippine President Ferdinand Marcos Marcos' preferred number is not 8, but 7.

So someone said to him, "Mr. President, don't you like 7 a lot?" Get out your calculator and I can send you all 7s. ”

Then, the man multiplied the "missing 8 number" by 63, and suddenly, the 777777777 met Mr. Marcos's eyes.

"Missing 8 numbers" is actually not a soft spot for 7, it is a bowl of water, and all numbers are treated equally:

All you need to do is use multiples of 9 (9, 18...... Until 81) to multiply it, then 111111111, 222222222...... Until 999999999 will appear one after another.

12345679×9=111111111

12345679×18=222222222

12345679×27=333333333

12345679×36=444444444

12345679×45=555555555

12345679×54=666666666

12345679×63=777777777

12345679×72=888888888

12345679×81=999999999

Two, the Trinity

"Missing 8 numbers" aroused the strong interest of researchers, so people continued to multiply it by multiples of 3, and found that the product was repeated in a "trinity".

12345679×12=148148148

12345679×15=185185185

12345679×21=259259259

12345679×30=370370370

12345679×33=407407407

12345679×36=444444444

12345679×42=518518518

12345679×48=592592592

12345679×51=629629629

12345679×57=703703703

12345679×78=962962962

12345679×81=999999999

The nine-digit number here is made up of "trinity" numbers, which is amazing!

3. Take turns to "rest"

When the multiplier is not a multiple of 3, there is no "one-size-fits-all" or "trinity" phenomenon, but a singular property can still be seen:

None of the digits of the product are identical. There is a clear pattern of what is missing, and they appear according to an "even distribution".

In addition, in the product, there is no zài in the case of missing 3, 6, and 9.

Let's start with a single digit scenario:

12345679×1=12345679 (0 and 8 missing)

12345679×2=24691358 (0 and 7 missing)

12345679×4=49382716 (0 and 5 missing)

12345679×5=61728395 (0 and 4 missing)

12345679×7=86419753 (0 and 2 missing)

12345679×8=98765432 (0 and 1 missing)

In the product above, the numbers 3, 6, and 9 are not missing, but 0 is missing. The other missing numbers are 8, 7, 5, 4, 2, 1, and they appear from highest to smallest.

Let's take a look at the situation where the multiplier is in the interval [10~17], where 12 and 15 are excluded because they are multiples of 3.

12345679×10=123456790 (missing 8)

12345679×11=135802469 (missing 7)

12345679×13=160493827 (missing 5)

12345679×14=172869506 (missing 4)

12345679×16=197530864 (missing 2)

12345679×17=209876543 (missing 1)

There are still 3, 6, and 9 missing from the above product, but there is no more 0, and the other number missing is similar to the previous one—once in order of magnitude.

It's so interesting to see what is missing in the product, just like the "rotational rest" of workers in a factory or store, where everyone has a share, but they can't eat more than they eat!

The multiplier is exactly the same in the [19~26] and other intervals (interval length equal to 7).

12345679×19=234567901 (missing 8)

12345679×20=246913580 (missing 7)

12345679×22=271604938 (missing 5)

12345679×23=283950617(missing 4)

12345679×25=308641975 (missing 2)

12345679×26=320987654 (missing 1)

Consistently, when the multiplier exceeds 81, the product will be at least ten digits, but the above phenomena still exist. Let's look at a few more examples:

(1) The multiplier is a multiple of 9

12345679×243=2999999997, as long as the leftmost number 2 of the product is added to the rightmost 7, it will still appear "all the same".

Another example: 12345679×108=1333333332 (the leftmost number 1 of the product is added to the rightmost 2, which is exactly equal to 3)

12345679×117=1444444443 (the leftmost number 1 of the product is added to the rightmost 3, which is exactly equal to 4)

12345679×171=2111111109 (the leftmost number of the product, 2, plus the rightmost "09", results in 11)

(2) The multiplier is a multiple of 3, but not a multiple of 9

12345679×84=1037037036, as long as you add the leftmost number 1 to the rightmost 6 of the product, you can see the "trinity" phenomenon.

(3) The multiplier is 3k+1 or 3k+2 type

12345679×98=1209876542, on the surface, there is a similar 2 in the product;

However, as mentioned above, if you add the leftmost number 1 of the product to the rightmost 2, the resulting number is 209876543, which is the number "missing 1".

According to the above "doctrine", it can be seen that it is the turn of 1 to rest at this time, and the result is completely consistent with the theory.

Fourth, the marquee

Winter goes to spring, and the 24 solar terms are still the beginning of spring, rain, and sting...... The order is completely unchanged, and the table xiàn is a periodic repetition.

The "missing 8" also has this property, but its multiplier is quite bizarre.

In fact, when the multiplier is 19, the product will be 234567901, and like a marquee, the number 2, which was originally in second place, becomes the trailbreaker.

In-depth research has shown that when the multiplier is an arithmetic progression with a tolerance equal to 9, the phenomenon of "marqueeping" occurs.

Now, let's change the multiplier to 10, 19, 28, 37, 46, 55, 64, 73 (they form a series of equal differences with a tolerance of 9):

12345679×10=123456790

12345679×19=234567901

12345679×28=345679012

12345679×37=456790123

12345679×46=567901234

12345679×55=679012345

12345679×64=790123456

12345679×73=901234567

The above products are all "missing 8 numbers"! The numbers 1, 2, 3, 4, 5, 6, 7, and 9 appear in turn on each digit like a marquee.

Fifth, palindrome pairs walk hand in hand

The "fine structure" of the "missing 8 numbers" aroused great interest among researchers, and people accidentally noticed that:

12345679×4=49382716

12345679×5=61728395

Isn't the product of the previous formula read upside down (from right to left) exactly the product of the latter formula?

(But there is a slight difference, i.e., 5 generations to 4, and according to the "doctrine of rotational rest", this is exactly what the question should be.) ）

The same should be true for pairs of multipliers such as 13, 14, 31, 32, etc. (the corresponding tolerance for each of the two adjacent pairs of multipliers is equal to 9).

For example:

12345679×13=160493827

12345679×14=172839506

12345679×22=271604938

12345679×23=283950617

12345679×67=827160493

12345679×68=839506172

Sixth, genetic factors

"Lack of 8 numbers" can also "have children", and these descendants inherit their "genetic factors" and completely inherit these above characteristics.

So almost all the members of this large family have the same skills as their ancestors 12345679.

For example, 506172839 is the product of "missing 8" and 41, so it is a derivative.

We see that 506172839×3=1518518517.

Add the leftmost number 1 of the product to the rightmost 7 to get 8. As mentioned earlier, the "Trinity" model is back to us.

"Missing 8 numbers" has an even more magical and spectacular palindromic phenomenon. Let's move on to multiplication:

12345679×9=111111111

12345679×99=1222222221

12345679×999=12333333321

12345679×9999=123444444321

12345679×99999=1234555554321

12345679×999999=12345666654321

12345679×9999999=123456777654321

12345679×99999999=1234567887654321

12345679×999999999=12345678987654321

A miracle happened! To the right of the equal sign are all palindromic numbers (read from left to right or right to left, the same number).

Moreover, these palindrome numbers are all "step-by-step" ascent and descent, magical, beautiful, and interesting!

Because 12345679 = 333667×37, the "missing 8 number" is a composite number.

There is a peculiar relationship between the "missing 8 number" and its two factors, 333667 and 37.

The first and last two numbers of a factor 333667, 3 and 7, form another factor 37;

And the sum of the numbers of "missing 8 numbers" itself, 1+2+3+4+5+6+7+9, is also equal to 37.

It can be seen that the "lack of 8 numbers" is naturally related to 37.

What's even more amazing is that 1/81 is converted into a decimal, and this decimal is also "missing 8 numbers":

1/81=0.012345679012345679012345679……

Why is there no shortage of other numbers, but 8?

It turns out that 1/81=1/9×1/9=0.1111...×0.11111....

Here 0.1111... is an infinitesimal with an infinite number of 1s after the decimal point.

The wonderful nature of the "missing 8" is epitomized by the large number of mathematical cycles, which are surprisingly regular.

The peculiar nature of the "lack of 8 numbers" has long aroused people's strong interest. And how many mysteries are in it, people will definitely uncover them all.

"Missing 8 numbers" is so amazing that I, who has no interest in mathematics, can't help but praise it!