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There are specific rules to remember divisibility by numbers up to 10.
1: All numbers 2: Even numbers 3: Sum of digits = 3 4: Divisible by 2 twice/Last two digits divisible by 4 5: Last digit = 5 or 0 6: Divisible by 3 and 2 7: No divisibility rule 8: Divisible by 2 thrice/last three digits divisible by 8 9: Sum of digits = 9 10: Last digit = 0
Now for bigger numbers, what you basically need to do is assume they are divisible and check for consistency with constrains given in the problem.
For example if a question asks if a number x is divisible by 12 and gives you additional constraints say:
x = 12a
And then check against restrictions. But I'd say that beyond this the method would be case specific. Hope this helps!