Definition of Divisibility:

①If a|b, a|c, then a|(b±c).

②If a|b, then for any c (c≠0), a|bc.

③ For any non-zero integer a, ±a|a=±1.

④If a|b, b|a, then |a|=|b|.

⑤a|b,a|c⇔∀x,y∈Z ,a|(b×x+c×y)

⑥If a is divisible by both b and c, and b and c are relatively prime, then a must be divisible by the product bc, and vice versa.

⑦If m≠0, then a|b⇔(m⋅a)|(m⋅b)If m≠0, then a|b⇔(m·a)|(m·b)

⑧x,y∈Z, and ∃a,b make ax+by=1,a|n,b|n⇒(a⋅b)|nx,y∈Z, and ∃a,b make ax+by=1,a|n,b|n⇒(a·b)|n

Numbers divisible by 2: multiples of 2 in the ones place
Numbers divisible by 4:The two-digit number consisting of ones and tens is divisible by 4
Numbers divisible by 8:The three-digit number consisting of hundreds, ones, and tens is divisible by 8 (note the order)
Numbers that are divisible by 5:A multiple of 5 in the ones digit
Numbers that are divisible by 25:The two-digit number formed by the tens and ones digit is divisible by 25
Numbers that are divisible by 125:Numbers consisting of hundreds, tens, and ones are divisible by 125
Numbers divisible by 3:The numbers on each digit and numbers divisible by 3
Numbers divisible by 9:The sum of the numbers on each digit is divisible by 9
Numbers that are divisible by 11:The number on odd digits (counting from left to right)The absolute value of the difference between the sum of numbers and the sum of numbers in even digits is divisible by 11
Numbers that are divisible by 7:Truncate the one digit, get a new number, and then subtract 2 times the single digit of the original number from the number. If the difference is a multiple of 7, the original number can be divisible by 7. If the judgment is not good, recurse processing
Numbers divisible by 13:Truncate the one digit to get a new number, then add 4 times the one digit of the original number to the number,If the sum is a multiple of 13, the original number is divisible by 13
Numbers that are divisible by 17:Truncate the one digit to get a newIf the sum is a multiple of 17, the original number can be divisible by 17, or the last three digits are truncated, and the remaining number is 3 timesIf the absolute value of the three-digit difference is divisible by 17, the prime number is divisible by 17
Numbers divisible by 19:Truncate the ones digit to get a new number, then add 2 times the one digit of the original number to the number, if the sum isIf it is a multiple of 19, the original number can be divisible by 19, or the last three digits can be truncated. If the absolute value of the difference between the remaining number and the last three digits of 7 times can be divisible by 19, then the original number can be divisible by 19strong>