GMAT number properties strategies

Test with concrete numbers. Number properties questions ask whether a statement is always true, sometimes true, or never true. Reasoning abstractly is slow and error-prone. Picking specific numbers and testing the statement against each one is faster and more reliable. The key is choosing numbers that cover the boundary cases, not just the easy ones.

These five numbers catch almost every edge case the GMAT uses. Zero is even but not positive. One is neither prime nor composite. Negative one flips the sign when squared changes the parity. Two is the only even prime. Negative two tests whether a property depends on positivity. If a statement holds for all five, it is likely always true. If it fails for any one, you have your counterexample. Add a fraction like one-half when the question allows non-integers.

If $p$ is a prime number greater than 2, then $p^2 - 1$ must be divisible by which of the following? Step 1: Identify the property. $p$ is an odd prime (all primes greater than 2 are odd), and we need to find what $p^2 - 1$ is always divisible by. Step 2: Test with concrete primes. Let $p = 3$: $3^2 - 1 = 8$. Let $p = 5$: $5^2 - 1 = 24$. Let $p = 7$: $7^2 - 1 = 48$. Let $p = 11$: $11^2 - 1 = 120$. Step 3: Find the common divisor. The values are 8, 24, 48, and 120. The greatest common divisor of 8 and 24 is 8. Check 48: divisible by 8. Check 120: divisible by 8. All are divisible by 8. Step 4: Why it works. An odd number squared is odd, so $p^2 - 1$ is even. More precisely, $p^2 - 1 = (p - 1)(p + 1)$. Since $p$ is odd, both $p - 1$ and $p + 1$ are consecutive even numbers, and one of any two consecutive even numbers is divisible by 4. So the product is divisible by 8. The answer is 8.