How can divisibility rules identify multiples of 2, 3, 5, 9, and 11?

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In number theory, divisibility rules provide efficient methods to determine if a number is a multiple of another without performing full division. Based on the theory of divisors, here are the rules for the requested integers:

2: The number is even (it ends in 0, 2, 4, 6, or 8).
3: The sum of the digits of the number is divisible by 3.
5: The number ends in either 0 or 5.
9: The sum of the digits of the number is divisible by 9.
11: The difference between the sum of the digits in the odd-numbered positions and the sum of the digits in the even-numbered positions is either 0 or a number divisible by 11.

These rules are useful tools in arithmetic for identifying factors and multiples within the fundamental theory of numbers.1