Why can a number whose sum is a multiple of 3 be divisible by 3? Why? There's a reason

Why can a number whose sum is a multiple of 3 be divisible by 3? Why? There's a reason

Let s have n bits, ana (n-1)... A2a1, and a1 + A2 +... + an = 3k, (k is a natural number) s - (a1 + A2 +... + an) = an * 10 ^ (n-1) +... + 10a2 + A1 - (a1 + A2 +... + an) = an * 99.9 +. + 99a3 + 9A2, then s = an * 99.9 +. + 99a3 + 9A2 + (a1 + A2 +... + an) is obviously divisible by 3

It is proved that if a two digit ten digit number and one digit number exchange positions, the new number and the original number will be divisible by 11

Let the two digit ten digit number be a and the one digit number be B
Then the two digits are 10A + B
After swapping positions, it's 10B + a
10A + B + 10B + a = 11a + 11b = 11 (a + b)
It's divisible by 11

The law of numbers being divisible
For example:
Numbers with 5 or 0 bits can be divided by "5"
All digits that add up to a multiple of 3 can be divisible by "3"
(excuse me, how can you tell whether other numbers 1-9 can be divided?

(1) The characteristics of 1 and 0 are as follows
1 is the divisor of any integer, that is, for any integer a, there is always 1|a
If 0 is a multiple of any non-zero integer, a ≠ 0 and a is an integer, then a | 0
(2) If the last bit of an integer is 0, 2, 4, 6 or 8, then the number can be divided by 2
(3) If the sum of the numbers of an integer can be divided by 3, then the integer can be divided by 3
(4) If the last two digits of an integer can be divided by 4, then the number can be divided by 4
(5) If the last bit of an integer is 0 or 5, the number can be divided by 5
(6) If an integer can be divided by 2 and 3, then the number can be divided by 6
(7) If the number of digits of an integer is truncated, and then 2 times of the number of digits is subtracted from the remaining number, if the difference is a multiple of 7, then the original number can be divided by 7. If the difference is too large or it is difficult to see whether it is a multiple of 7 by mental arithmetic, the above process of "truncation, multiplication, subtraction and difference checking" needs to be continued until it can be clearly judged, The process of judging whether 133 is a multiple of 7 is as follows: 13-3 × 2 = 7, so 133 is a multiple of 7; for example, the process of judging whether 6139 is a multiple of 7 is as follows: 613-9 × 2 = 595, 59-5 × 2 = 49, so 6139 is a multiple of 7, and so on
(8) If the last three digits of an integer can be divided by 8, then the number can be divided by 8
(9) If the number sum of an integer can be divided by 9, then the integer can be divided by 9
(10) If the last bit of an integer is 0, the number can be divided by 10
(11) If the difference between the sum of odd digits and the sum of even digits of an integer can be divided by 11, then the number can be divided by 11. The multiple test method of 11 can also be processed by the "tail cutting method" of check 7. The only difference in the process is that the multiple is not 2, but 1!
(12) If an integer can be divided by 3 and 4, then the number can be divided by 12
(13) If the number of one digit of an integer is truncated, and then four times of the number of one digit is added to the remaining number, if the difference is a multiple of 13, then the original number can be divided by 13. If the difference is too large or it is difficult to see whether it is a multiple of 13 by mental arithmetic, we need to continue the above process of "truncation, multiplication, addition and difference checking" until we can make a clear judgment
(14) If the number of one digit of an integer is truncated, and then five times of the number of one digit is subtracted from the remaining number, if the difference is a multiple of 17, then the original number can be divided by 17. If the difference is too large or it is difficult to see whether it is a multiple of 17 by mental arithmetic, we need to continue the above process of "truncation, multiplication, subtraction and difference checking" until we can make a clear judgment
(15) If the number of one digit of an integer is truncated, and then two times of the number of one digit is added to the remaining number, if the difference is a multiple of 19, then the original number can be divided by 19. If the difference is too large or it is difficult to see whether it is a multiple of 19 by mental arithmetic, we need to continue the above process of "truncation, multiplication, addition and difference checking" until we can make a clear judgment
(16) If the difference between the last three digits of an integer and the preceding number of 3 times can be divided by 17, then the number can be divided by 17
(17) If the difference between the last three digits of an integer and the preceding number of 7 times can be divided by 19, then the number can be divided by 19
(18) If the difference between the last four digits of an integer and the first five times of the number separated can be divided by 23 (or 29), then the number can be divided by 23
Someone answered that before. Take a look at the resources link below