How to Solve: Divisibility Rules (GRE/GMAT) – Complete Guide

How to Solve: Divisibility Rules (GRE/GMAT) – Complete Guide

Score Impact: This question type appears 3-5 times per GRE/GMAT Quant section. Mastering it saves 2-3 minutes per test and boosts your score by 50+ points—enough to move from the 70th to the 90th percentile.

WHAT THIS QUESTION TYPE IS ACTUALLY TESTING

The exam isn’t testing your ability to divide numbers. It’s probing for: 1. Pattern recognition – Can you spot divisibility rules without brute-force division? 2. Logical elimination – Can you quickly discard wrong answers using constraints? 3. Attention to traps – Do you fall for "almost correct" distractors (e.g., numbers divisible by 2 but not 4)?

ANATOMY OF THE QUESTION

Structure Breakdown

1. Stem: A number (or variable expression) is given, often with conditions (e.g., "If n is divisible by 6…").
2. Conditions: Additional constraints (e.g., "and n is odd," "and n > 100").
3. Answer Choices: 4-5 options, usually numbers or expressions. One is correct; others are traps.
4. What to Ignore: Irrelevant details (e.g., "the number is written on a blue card").

Representative Example

If n is an integer divisible by 12, which of the following must also be divisible by 12? (A) n + 6 (B) n + 12 (C) 2n (D) n/2 (E) n + 24

THE DECISION FRAMEWORK (Step-by-Step)

Run this process every time. No exceptions.

1. Identify the divisor – What number are we checking divisibility for? (e.g., 12).
2. Factor the divisor – Break it into primes (e.g., 12 = 2² × 3).
3. Apply the rule – For a number to be divisible by d, it must include all prime factors of d in at least the same powers.
4. Test each answer choice – Use the rule to eliminate options that fail.
5. Check for traps – Look for "off-by-one" errors (e.g., divisible by 6 but not 12).

Worked Examples

Example 1 – Straightforward

If x is divisible by 8, which of the following must be divisible by 4? (A) x + 4 (B) x + 8 (C) x/2 (D) 3x (E) x – 4

Step-by-Step: 1. Divisor: 8 (for x), but we’re checking for divisibility by 4. 2. Factor: 8 = 2³, 4 = 2². 3. Rule: Since x is divisible by 8, it has at least three 2’s. Any multiple of x or x ± a multiple of 8 will retain at least two 2’s. 4. Test choices:
- (A) x + 4: x is divisible by 8, so x = 8k. Then x + 4 = 8k + 4 = 4(2k + 1) → divisible by 4. Keep.
- (B) x + 8: x + 8 = 8k + 8 = 8(k + 1) → divisible by 8 (and thus 4). Keep.
- (C) x/2: x = 8k → x/2 = 4k → divisible by 4. Keep.
- (D) 3x: x = 8k → 3x = 24k → divisible by 8 (and thus 4). Keep.
- (E) x – 4: x – 4 = 8k – 4 = 4(2k – 1) → divisible by 4. Keep. 5. Trap check: All options work! But the question asks for must be divisible by 4. Since all pass, the answer is all of the above—but this is rare. Re-examine:
- (C) x/2: If x = 8, x/2 = 4 (divisible by 4). If x = 16, x/2 = 8 (divisible by 4). Always true.
- Answer: (D) 3x is the safest (always divisible by 8, hence 4).

Elimination Logic: - (A), (B), (E) work but are less general than (D). - (C) fails if x = 0 (but 0 is divisible by 4, so it still works).

Example 2 – Common Trap Version

If n is divisible by 6, which of the following must be divisible by 3? (A) n + 3 (B) n + 6 (C) n/2 (D) 2n (E) n + 1

Step-by-Step: 1. Divisor: 6 (for n), but we’re checking for 3. 2. Factor: 6 = 2 × 3. 3. Rule: n is divisible by 6 → n = 6k = 2 × 3 × k. For divisibility by 3, the number must include at least one 3. 4. Test choices:
- (A) n + 3: n = 6k → n + 3 = 6k + 3 = 3(2k + 1) → divisible by 3. Keep.
- (B) n + 6: n + 6 = 6k + 6 = 6(k + 1) → divisible by 3. Keep.
- (C) n/2: n = 6k → n/2 = 3k → divisible by 3. Keep.
- (D) 2n: 2n = 12k → divisible by 3. Keep.
- (E) n + 1: n = 6k → n + 1 = 6k + 1 → not divisible by 3. Eliminate. 5. Trap check: (C) n/2 fails if n = 6 (6/2 = 3, which is divisible by 3). But if n = 12, 12/2 = 6 (divisible by 3). Always true.
- However, (A) and (B) are also always true. The question asks for must—so multiple answers are possible. But (D) 2n is the most general.
- Answer: (D) 2n.

Why the Trap? - (E) is the classic "off-by-one" distractor. - (C) seems risky (what if n is odd?), but n is divisible by 6 → n is even → n/2 is integer.

Example 3 – Hard Variant

If k is an integer divisible by 18, which of the following must be divisible by 27? (A) k + 9 (B) k + 18 (C) k + 27 (D) 2k (E) k/3

Step-by-Step: 1. Divisor: 18 (for k), but we’re checking for 27. 2. Factor: 18 = 2 × 3², 27 = 3³. 3. Rule: k is divisible by 18 → k = 18m = 2 × 3² × m. For divisibility by 27, the number must have at least three 3’s. 4. Test choices:
- (A) k + 9: k = 18m → k + 9 = 18m + 9 = 9(2m + 1). This has two 3’s → not divisible by 27. Eliminate.
- (B) k + 18: k + 18 = 18m + 18 = 18(m + 1). Still only two 3’s. Eliminate.
- (C) k + 27: k + 27 = 18m + 27 = 9(2m + 3). Two 3’s → Eliminate.
- (D) 2k: 2k = 36m = 2² × 3² × m. Only two 3’s → Eliminate.
- (E) k/3: k = 18m → k/3 = 6m = 2 × 3 × m. Only one 3 → Eliminate. 5. Wait! None work. Did we miss something?
- Re-examine (E): k/3 = 6m. But k is divisible by 18 → m must be divisible by 3 (since 18m/3 = 6m → m = 3n). So k/3 = 6 × 3n = 18n → 2 × 3² × n. Still only two 3’s.
- Conclusion: The question is flawed, or we need to assume m is divisible by 3. But the GMAT/GRE would never do this.
- Alternative approach: Pick numbers. Let k = 18 (divisible by 18).
- (A) 18 + 9 = 27 → divisible by 27. But wait! k = 36 → 36 + 9 = 45 (not divisible by 27). So (A) fails.
- (B) 18 + 18 = 36 → not divisible by 27. Fails.
- (C) 18 + 27 = 45 → not divisible by 27. Fails.
- (D) 218 = 36 → not divisible by 27. Fails.
- (E) 18/3 = 6 → not divisible by 27. Fails.
- Answer: None of the above. But since this is a test question, the correct answer is likely (C) k + 27 if k is a multiple of 54 (e.g., k = 54 → 54 + 27 = 81, divisible by 27). But the question says "must," so this is a trick question.

Key Insight: - The question is testing whether you assume k has exactly two 3’s. If k has three 3’s (e.g., k = 54), then (C) works. But the question says "divisible by 18," not "exactly divisible by 18." - Answer: (C) k + 27 (assuming k is a multiple of 54).

WRONG ANSWER PATTERNS

1. Off-by-one error → n + 6 is divisible by 6, but n + 3 is not.
2. Why it looks right: "Adding a multiple of 6 keeps it divisible by 6."

3. Why it’s wrong: n + 3 may not retain the factor of 2 (e.g., n = 6 → 6 + 3 = 9, not divisible by 6).

4. Ignoring prime factors → Assuming n divisible by 6 means n + 6 is divisible by 12.

5. Why it looks right: "6 is a factor of 12."

6. Why it’s wrong: n + 6 may not have the required two 2’s (e.g., n = 6 → 6 + 6 = 12, which works; but n = 12 → 12 + 6 = 18, not divisible by 12).

7. Division trap → n/2 is divisible by 3 if n is divisible by 6.

8. Why it looks right: "6/2 = 3, which is divisible by 3."

9. Why it’s wrong: n = 6 → 6/2 = 3 (works); but n = 12 → 12/2 = 6 (works). Actually correct, but students doubt it.

10. Overgeneralizing → Assuming n + d is divisible by d if n is divisible by d.

11. Why it looks right: "Adding a multiple of d keeps it divisible by d."
12. Why it’s wrong: True for n + d, but not for n + k where k ≠ d.

Common Mistakes

1. Mistake: Forgetting to factor the divisor.
2. Why it happens: Rushing to test numbers without breaking down primes.

3. Correct approach: Always factor the divisor first (e.g., 12 = 2² × 3).

4. Mistake: Testing only one number.

5. Why it happens: Confirmation bias ("It worked for n = 6, so it must be true").

6. Correct approach: Test at least two numbers (e.g., n = 6 and n = 12).

7. Mistake: Ignoring "must" vs. "could."

8. Why it happens: Assuming an option that works for some n is always correct.

9. Correct approach: Only pick options that work for all valid n.

10. Mistake: Misapplying the rule for subtraction.

11. Why it happens: Assuming n – d is divisible by d if n is divisible by d.

12. Correct approach: n – d = d(k – 1), which is divisible by d. This is correct, but students doubt it.

13. Mistake: Confusing divisibility with prime factors.

14. Why it happens: Thinking "divisible by 6" means "has a 6 in its factors."
15. Correct approach: "Divisible by 6" means "has at least one 2 and one 3."

TIME STRATEGY

• Target time: 45–60 seconds per question.
• When to skip: If you can’t factor the divisor in 10 seconds, flag and return.
• Minimum work:
• Factor the divisor.
• Test one number (e.g., n = smallest multiple of the divisor).
• Eliminate options that fail.

BACKSOLVING AND SHORTCUTS

1. Pick numbers: Choose the smallest multiple of the divisor (e.g., for divisibility by 12, pick n = 12).
2. Eliminate first: Cross out options that don’t work for your test number.
3. Check for "must": If an option works for n = 12 but fails for n = 24, it’s wrong.
4. Prime factor shortcut: If the divisor is d = p^a × q^b, the number must have at least p^a and q^b.

1-Minute Recap

"Here’s the exact process to solve any divisibility question in under a minute: 1. Factor the divisor – Break it into primes. 12? 2² × 3. 18? 2 × 3². 2. Test one number – Pick the smallest multiple of the divisor (e.g., 12 for divisibility by 12). 3. Plug and eliminate – Test each answer choice. If it fails for your number, cross it out. 4. Check for traps – Does the option work for all multiples, or just your test number? If not, eliminate. 5. Confirm "must" – The correct answer must work for every valid input.

Example: If n is divisible by 8, which must be divisible by 4? - 8 = 2³, 4 = 2². - Test n = 8: - n + 4 = 12 → divisible by 4. Keep. - n + 8 = 16 → divisible by 4. Keep. - n/2 = 4 → divisible by 4. Keep. - 2n = 16 → divisible by 4. Best answer. - Answer: 2n.

Now go practice—this is free points on test day."

Final Note: Every line above is actionable under timed conditions. Memorize the framework, drill the examples, and you’ll never miss a divisibility question again.