These can be derived by taking successive powers of 10 modulo the number you wish to create a rule for.
A full explanation will have to wait for another day.
Ways of telling whether an integer is divisible by:
Drop an apple. If it falls downwards, the integer is divisible by 1.
Yes, if the last digit is divisible by two.
Yes, if the sum of the digits is divisible by 3.
Yes, if the last digit of the integer divided by two is divisible by two.
Yes, if the last digit is a 0 or a 5.
Divide the integer by 2. Yes, if the sum of the digits of the result is divisible by three.
Let the digits of the number be where has the lowest place value. The number is divisible by 7 if 7 divides the sum of
Example: Does 7 divide 3236655
Answer: yes, since
and as 7 clearly divides 28, 7 divides 91, hence 7 divides 3236655.
Yes, if the sum of the digits is divisible by 9.