GMAT arithmetic strategies

Test cases systematically. When a question asks whether a statement is always true, do not reason abstractly. Pick concrete numbers that cover the full range of possibilities and test each one. If the question involves divisibility, test a multiple of the divisor, a non-multiple, and a boundary value. Systematic testing is faster and more reliable than trying to prove a rule in your head.

A number is divisible by 3 if the sum of its digits is divisible by 3. The same rule works for 9. A number is divisible by 4 if its last two digits form a number divisible by 4. A number is divisible by 6 if it is divisible by both 2 and 3. These rules let you skip long division. For example, 471 is divisible by 3 because 4 plus 7 plus 1 is 12, and 12 is divisible by 3. It is not divisible by 9 because 12 is not divisible by 9.

If $n$ is a positive integer and $n^2$ is divisible by 12, then $n$ must be divisible by which of the following? Step 1: Identify the property. The question tests divisibility, and asks what $n$ must be divisible by. Step 2: Pick numbers. The smallest $n$ where $n^2$ is divisible by 12 is $n = 6$, because $6^2 = 36$, and 36 is divisible by 12. The next is $n = 12$, because $12^2 = 144$, and 144 is divisible by 12. Step 3: Test each against the answer choices. For $n = 6$: divisible by 2, divisible by 3, divisible by 6, not divisible by 4, not divisible by 12. For $n = 12$: divisible by all of those. Step 4: What must be true for every valid $n$? Both 6 and 12 are divisible by 6. The answer is 6.