Every number is divisible by any non-zero number. Any element of the set of numbers of the form 4518*k where k is an integer is evenly divisible.
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The lists of numbers divisible by and not divisible by 600 are both infinite.
All even numbers are divisible by 2.
Odd numbers
Yes, they must be.
Numbers for which the sum of the digits is divisible by 9. This is also true for 3. There are other divisibility/multiple tests for other numbers (e.g., numbers that are divisible by 5 end in 5 or 0; numbers whose last two digits are divisible by 4 are divisible by 4)