The nature of divisibility: (1) If A is divisible by B and C is an arbitrary integer, then the product ac can also be divisible by B; (2) If A is divisible by both B and C, and B and C are coprime, A must be divisible by the product bc, and vice versa.
Rule 1: Any integer can be divisible by 1
Note: The following is the decimal representation of integers.
Article 2: Numbers in units of 2, 4, 6, 8 and 0 can be divisible by 2. [2]
Article 3: If the sum of the digits of each digit is divisible by 3, then this number can be divisible by 3.
Article 4: If the last two digits are divisible by 4, this number can be divisible by 4.
Article 5: Any number in units of 0 or 5 can be divisible by 5.
Article 6: As long as a number is divisible by both 2 and 3, it can be divisible by 6.
Article 7: Cut off the single digits, and then subtract twice the single digits from the remainder. If the difference is a multiple of 7, the original number can be divisible by 7.
Article 8: If the last three digits are divisible by 8, this number can be divisible by 8.
Article 9: If the sum of digits of each number is divisible by 9, then this number can be divisible by 9.
Article 10: If the last digit of an integer is 0, this number can be divisible by 10.
Article 11: Count a number from right to left, add the odd and even numbers respectively, and then subtract the sum of the two numbers. If the difference is divisible by 1 1 (including the difference is 0), the original number can be divisible by 1 1.
Article 12: If an integer is divisible by 3 and 4, it is divisible by 12.
Article 13: If the single digits of an integer are truncated, then four times the single digits are added to the remainder. If the sum is a multiple of 13, the original number can be divisible by 13. If the difference is too big or it is not easy to see whether it is a multiple of 13 in mental arithmetic, you need to continue the above process until you can clearly judge.
Article 14 If the single digit of an integer A is truncated, 5 times of the single digit shall be subtracted from the remainder. If the difference is a multiple of 17, the original number can be divisible by 17. If the difference is too big or it is not easy to see whether it is a multiple of 17 in mental arithmetic, you need to continue the above process until you can clearly judge. B. If the difference between the last three digits of an integer and the three times of the previous separated number is divisible by 17, then this number can be divisible by 17.
Article 15 a If the single digit of an integer is truncated, then twice the single digit is added to the remainder. If the sum is a multiple of 19, the original number can be divisible by 19. If the difference is too big or it is not easy to see whether it is a multiple of 19 in mental arithmetic, you need to continue the above process until you can clearly judge. B. If the difference between the last three digits of an integer and the previous partition number is 7 times divisible by 19, then this number can be divisible by 19.
Article 16: If the difference between the last four digits of an integer and the first five times of the separated number can be divisible by 23, then this number can be divisible by 23.
Article 17: If the difference between the last four digits of an integer and the first five times of the separated number can be divisible by 29, then this number can be divisible by 29.
Article 18: If the difference between the last four digits of an integer and the previous number is divisible by 73, then this number is divisible by 73.
Article 19: If the difference between the last four digits of an integer and the previous number is divisible by 137, then this number can be divisible by 137.
Article 20: If the difference between the last four digits of an integer and the first five times of the separated number can be divisible by 23 (or 29), then this number can be divisible by 23.
Article 21: If the difference between the last five digits of an integer and the previous number is divisible by 909 1, then this number can be divisible by 909 1.
Article 22: If the sum of several segments of an integer is divisible by 9, then the number can be divisible by 9.
Article 23: Divide an integer into several segments, and add an odd number at the end of each segment and subtract an even number. If the result is divisible by 1 1, then this number can be divisible by 1 1.
Article 24: (1) If the sum of the last four digits of an integer and the previous number is divisible by 10 1, then this number can be divisible by10/.
(b) If the difference between the last two digits of an integer and the previous number is divisible by 10 1, then this number can be divisible by 10 1. For example, Article 7 (7) of the law of divisibility: cut off the single digits, and then subtract twice the single digits from the remainder. If the difference is a multiple of 7, the original number can be divisible by 7.
Example: ① 147 is 14 after single digit truncation, and 14-7*2=0 is a multiple of 7, so 147 is also a multiple of 7.
②2 198, if one digit is cut off, it is 2 19, 219-8 * 2 = 203; Go on, after removing the single digits, it is 20,20-3 * 2 = 14, and14 is a multiple of 7, so 2 198 is also a multiple of 7.
Attribute (1) If a|b and b|c, then a | c.
(2) if a|b, then a|kb (where k is an integer)
(3) if a|bc is coprime with a and c, then a | b.
(4) if a|b, a|c, then a | (b c)
(5) If b|a and c|a are coprime, the divisibility of bc|a has the following basic properties:
(1) If a|b, a|c, then a | (b c).
② if a|b, then any c, a|bc.
③ For any nonzero integer a, 1 | a, a | a.
④ If a|b and b|a, then |a|=|b|.
For any integer A, B, b>0, there is a unique number pair Q, R such that a=bq+r, where 0 ≤ R.
If c|a and c|b, then c is the common factor of a and b, if d is the common factor of a and b, d≥0 and is divisible by any common factor of a and b, then d is the greatest common factor of a and b. If the greatest common factor of a and b is equal to 1, then a and b are called coprime, also called coprime. The greatest common factor of a and b can be found repeatedly by division with remainder. This method is often called division by turns. Also known as Euclid algorithm.
I hope this helps.