First hunch: look at the last digit
A number ending in 5 is always divisible by 5. A number ending in 3 or 9 might be divisible by 3, but the last digit does not decide it.
The reliable tests
Factor of 5: the number ends in 0 or 5.
Factor of 3: add the digits. If the digit sum is divisible by 3, the number is divisible by 3.
Both 3 and 5: the number passes both tests, so it is divisible by 15.