What are the characteristics of integers divisible by 3, 4, 7, 8, 9, 11, 13 and 25?

What are the characteristics of integers divisible by 3, 4, 7, 8, 9, 11, 13 and 25?

The number divided by three must be a multiple of three, for example, 136,1 + 3 + 6 = 10, 147 = 1 + 4 + 7 = 12
If the last two digits of an integer can be divided by 4, then the number can be divided by 4
Divide by 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 calculation, 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
If the last three digits are divisible by 8, the number will be divisible by 8
If the sum of the digits of an integer can be divided by 9, then the integer can be divided by 9. For example, 252 = 2 + 2 + 5 = 9
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!
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
The last two digits are 00, 25, 50 and 75, which can be divided by 25