How to Solve: Number Properties (GRE/GMAT)

How to Solve: Number Properties (GRE/GMAT)

Score Impact: "Number Properties questions appear 4-6 times per GRE Quant section and 3-5 times per GMAT Quant section—mastering them can boost your score by 50+ points, moving you from the 60th to the 80th percentile."

WHAT THIS QUESTION TYPE IS ACTUALLY TESTING

The GRE/GMAT doesn’t test advanced math—it tests logical reasoning under constraints. Number Properties questions probe: 1. Precision in applying definitions (e.g., prime, integer, even/odd, factors/multiples) without overcomplicating. 2. Pattern recognition—spotting hidden relationships (e.g., divisibility rules, consecutive integers, remainders). 3. Elimination efficiency—using answer choices to reverse-engineer the solution when algebra stalls.

Trap: Students waste time on complex algebra when simple number substitution or divisibility rules would suffice.

ANATOMY OF THE QUESTION

Structure Breakdown

1. Stem: A short scenario (e.g., "If n is an integer, which of the following must be true?").
2. Conditions: Constraints on variables (e.g., n > 0, n is odd, n is a multiple of 3).
3. Answer Choices: 3-5 options, often mixing algebraic expressions, inequalities, or numerical values.
4. What to Ignore:
5. Overly complex expressions (e.g., n³ + 5n² – 2n + 1)—these are distractions.
6. Absolute values or exponents unless the stem explicitly mentions them.

Representative Example

Question: If x is an integer greater than 1, which of the following must be true? I. x² – x is even II. x³ + 1 is divisible by 3 III. x² + 1 is prime

(A) I only (B) II only (C) I and II only (D) I and III only (E) I, II, and III

THE DECISION FRAMEWORK (Step-by-Step)

Run this process for every Number Properties question:

1. Identify the variable’s constraints.
2. Is it an integer? Positive/negative? Even/odd? Prime?

3. Example: Here, x is an integer > 1.

4. Test small numbers first.

5. Plug in x = 2, 3, 4 (or x = –2, –1 if negatives are allowed).

6. Why? Small numbers reveal patterns faster than algebra.

7. Evaluate each statement separately.

8. For each Roman numeral, ask: "Does this hold for all valid values of x?"

9. If a statement fails for any value, eliminate it.

10. Eliminate answer choices.

11. Cross out options that include false statements.

12. Example: If II fails for x = 2, eliminate (B), (C), and (E).

13. Check edge cases.

14. Test x = 1 (if allowed), x = 0, or large numbers (e.g., x = 100).

15. Why? Some statements fail only at extremes.

16. Confirm the remaining choice.

17. If two options remain, test another number to break the tie.

Worked Examples

Example 1 – Straightforward

Question: If n is a positive integer, which of the following must be true? I. n² + n is even II. n³ – n is divisible by 6 III. n² + 1 is odd

(A) I only (B) II only (C) I and II only (D) I and III only (E) I, II, and III

Step-by-Step: 1. Constraints: n is a positive integer. 2. Test n = 1, 2, 3:
- I: 1² + 1 = 2 (even), 2² + 2 = 6 (even), 3² + 3 = 12 (even) → True.
- II: 1³ – 1 = 0 (divisible by 6), 2³ – 2 = 6 (divisible by 6), 3³ – 3 = 24 (divisible by 6) → True.
- III: 1² + 1 = 2 (even) → False (must be odd). 3. Eliminate: III is false → eliminate (D) and (E). 4. Check n = 4:
- II: 4³ – 4 = 60 (divisible by 6) → still true. 5. Answer: (C) I and II only.

Example 2 – Common Trap Version

Question: If k is an integer and k² – 4k + 3 is odd, which of the following must be true? (A) k is even (B) k is odd (C) k is a multiple of 3 (D) k + 1 is even (E) k – 1 is odd

Trap: Students assume k must be odd because k² is odd. But k² – 4k + 3 is odd, not k² alone.

Step-by-Step: 1. Constraints: k is an integer; k² – 4k + 3 is odd. 2. Test k = 0, 1, 2, 3:
- k = 0: 0 – 0 + 3 = 3 (odd) → k is even.
- k = 1: 1 – 4 + 3 = 0 (even) → invalid.
- k = 2: 4 – 8 + 3 = –1 (odd) → k is even.
- k = 3: 9 – 12 + 3 = 0 (even) → invalid. 3. Pattern: Only even k values work. 4. Check k = –2: 4 + 8 + 3 = 15 (odd) → k is even. 5. Eliminate:
- (B) k is odd → false.
- (C) k is a multiple of 3 → false (k = 2 works).
- (D) k + 1 is even → k = 0 → 1 (odd) → false.
- (E) k – 1 is odd → k = 2 → 1 (odd) → True, but k = 0 → –1 (odd) → True.
- Wait: (A) k is even → True for all valid k. 6. Answer: (A) k is even.

Why (E) is wrong: It’s sometimes true, but the question asks for must be true.

Example 3 – Hard Variant

Question: If p is a prime number greater than 3, which of the following must be divisible by 12? (A) p² – 1 (B) p² + 1 (C) p³ – p (D) p⁴ – 1 (E) p⁴ + p²

Step-by-Step: 1. Constraints: p is prime > 3 → p is odd and not divisible by 3. 2. Test p = 5, 7:
- A: 5² – 1 = 24 (divisible by 12), 7² – 1 = 48 (divisible by 12).
- B: 5² + 1 = 26 (not divisible by 12).
- C: 5³ – 5 = 120 (divisible by 12), 7³ – 7 = 336 (divisible by 12).
- D: 5⁴ – 1 = 624 (divisible by 12), 7⁴ – 1 = 2400 (divisible by 12).
- E: 5⁴ + 5² = 625 + 25 = 650 (not divisible by 12). 3. Eliminate: (B) and (E) are out. 4. Check divisibility by 3 and 4 (since 12 = 3 × 4):
- A: p² – 1 = (p – 1)(p + 1).
- p is odd → p – 1 and p + 1 are consecutive even numbers → one is divisible by 4.
- p is prime > 3 → p is not divisible by 3 → either p – 1 or p + 1 is divisible by 3.
- Thus, p² – 1 is divisible by 12.
- C: p³ – p = p(p² – 1).
- p² – 1 is divisible by 12 (from A), and p is odd → total divisibility by 12.
- D: p⁴ – 1 = (p² – 1)(p² + 1).
- p² – 1 is divisible by 12 (from A), and p² + 1 is even → total divisibility by 24. 5. Narrow down: (A), (C), and (D) work. But the question asks for must be divisible by 12.
- (A) is the simplest expression that always satisfies this. 6. Answer: (A).

WRONG ANSWER PATTERNS

1. The "Almost True" Distractor
2. Why it looks right: The statement holds for most numbers but fails at an edge case (e.g., n = 1).

3. Why it’s wrong: "Must be true" means all valid numbers, not "most."

4. The Overcomplication Trap

5. Why it looks right: The answer choice has complex algebra (e.g., n³ + 2n² – n).

6. Why it’s wrong: The GMAT/GRE favors simple patterns (e.g., n(n + 1) is always even).

7. The Reverse Logic Error

8. Why it looks right: The student assumes A → B implies B → A (e.g., "If n is even, n² is even" vs. "If n² is even, n is even").

9. Why it’s wrong: Only the second statement is always true.

10. The "Too Specific" Answer

11. Why it looks right: The answer choice includes a condition that’s true for some numbers but not all (e.g., "divisible by 6" when the question asks for "divisible by 4").
12. Why it’s wrong: It doesn’t satisfy the minimum requirement.

Common Mistakes

1. Mistake: Assuming all variables are positive.
2. Why it happens: Students forget to test negative numbers or zero.

3. Correct approach: Check the stem for constraints (e.g., "positive integer" vs. "integer").

4. Mistake: Skipping number substitution.

5. Why it happens: Over-reliance on algebra under time pressure.

6. Correct approach: Test n = 1, 2, 3 first—it’s faster than factoring.

7. Mistake: Misapplying divisibility rules.

8. Why it happens: Confusing "divisible by 6" with "divisible by 2 and 3 separately."

9. Correct approach: A number is divisible by 6 only if it’s divisible by both 2 and 3.

10. Mistake: Ignoring Roman numeral questions.

11. Why it happens: Students panic at the format and guess.

12. Correct approach: Evaluate each statement independently before looking at answer choices.

13. Mistake: Forgetting to test edge cases.

14. Why it happens: Testing n = 2, 3 but not n = 1 or n = 100.
15. Correct approach: Always test the smallest and largest possible values.

TIME STRATEGY

• Target time: 1:30–2:00 per question.
• When to skip:
• If you’re stuck after 2:30, flag and move on.
• If the question involves complex algebra (e.g., n⁴ – 1), skip to number substitution.
• Minimum work to answer confidently:
• Test 2–3 small numbers.
• Eliminate 2–3 answer choices.
• If stuck, guess from the remaining options.

BACKSOLVING AND SHORTCUTS

1. Plug in answer choices (backsolving):
2. For "which of the following must be true," test each Roman numeral with n = 1, 2, 3.

3. Example: If II fails for n = 2, eliminate all choices containing II.

4. Use divisibility shortcuts:

5. A number is divisible by 4 if its last two digits are divisible by 4.

6. A number is divisible by 9 if the sum of its digits is divisible by 9.

7. Even/odd rules:

8. Even ± Even = Even
9. Odd ± Odd = Even
10. Even ± Odd = Odd

11. Even × Any = Even

12. Prime number shortcuts:

13. All primes > 3 are of the form 6k ± 1 (e.g., 5, 7, 11, 13).

14. p² – 1 is always divisible by 24 for primes > 3.

15. Consecutive integers:

16. The product of n consecutive integers is divisible by n!.
17. Example: n(n + 1)(n + 2) is divisible by 6.

1-Minute Recap

"Here’s the exact process to crush Number Properties questions in under 2 minutes:

1. Read the stem carefully. Is the variable an integer? Positive? Even? Prime? These constraints are your lifeline.
2. Test small numbers first. Plug in n = 1, 2, 3 (or –1, 0 if allowed). If a statement fails for any number, eliminate it.
3. For Roman numerals, evaluate each one separately. Don’t look at the answer choices until you’ve checked all three.
4. Eliminate aggressively. If a statement is false for n = 2, cross out every answer choice that includes it.
5. Check edge cases. Test n = 1 and n = 100 to catch hidden traps.
6. If stuck, guess and move on. Spending 3+ minutes on one question costs you 2–3 correct answers later.

Remember: The GMAT/GRE doesn’t test math—it tests your ability to spot patterns under pressure. Master this framework, and you’ll gain 50+ points."

Final Tip: Drill 10–15 Number Properties questions daily for 2 weeks. Focus on speed + elimination, not perfection. You’ve got this!