**GMAT Focus Edition: Arithmetic – Integers, Factors, Multiples**

GMAT Focus Edition: Arithmetic – Integers, Factors, Multiples

Complete Study Guide for High-Scoring Candidates

What This Is

Integers, factors, and multiples form the backbone of GMAT arithmetic. These concepts appear in ~15% of Quant questions (Problem Solving and Data Sufficiency) and are foundational for higher-level topics like number properties, divisibility, and combinatorics. Mastering them ensures you avoid careless errors and solve problems efficiently—critical for breaking 700+.

Real-GMAT Example:
If ( n ) is a positive integer, is ( n^2 - n ) divisible by 6? (1) ( n ) is divisible by 3.
(2) ( n ) is odd.
Why it matters: This tests your ability to combine factor/multiple rules with logical reasoning—a hallmark of 700+ questions.

Key Concepts & Techniques

1. Prime Factorization
2. Break any integer into a product of primes (e.g., ( 60 = 2^2 \times 3 \times 5 )).

3. When to use: Finding LCM, GCF, or testing divisibility (e.g., "Is ( 12! + 1 ) divisible by 13?").

4. Greatest Common Factor (GCF) / Least Common Multiple (LCM)

5. GCF: Largest integer dividing two numbers (e.g., GCF of 12 and 18 is 6).
6. LCM: Smallest integer divisible by both (e.g., LCM of 4 and 6 is 12).

7. When to use: Word problems involving overlapping cycles (e.g., "Two lights flash every 12 and 18 seconds. When will they flash together?").

8. Divisibility Rules

9. 2: Last digit even.
10. 3: Sum of digits divisible by 3.
11. 4: Last two digits divisible by 4.
12. 5: Last digit 0 or 5.
13. 6: Divisible by 2 and 3.
14. 9: Sum of digits divisible by 9.

15. When to use: Quickly eliminating answer choices or testing sufficiency in DS.

16. Even/Odd Properties

17. Even ± Even = Even; Odd ± Odd = Even; Even ± Odd = Odd.
18. Even × Any = Even; Odd × Odd = Odd.

19. When to use: Questions like "If ( x ) is odd, is ( x^2 - 1 ) divisible by 8?" (Answer: Yes, because ( x^2 - 1 = (x-1)(x+1) ), two consecutive evens → divisible by 8).

20. Consecutive Integers

21. Any set of ( n ) consecutive integers contains a multiple of ( n ).

22. When to use: Proving divisibility (e.g., "Is ( n(n+1)(n+2) ) divisible by 6?" → Yes, because it includes multiples of 2 and 3).

23. Remainders

24. If ( a = bq + r ), then ( r ) is the remainder when ( a ) is divided by ( b ).

25. When to use: Questions like "What is the remainder when ( 7^{10} ) is divided by 5?" (Use cyclicity: ( 7^1 \equiv 2 ), ( 7^2 \equiv 4 ), ( 7^3 \equiv 3 ), ( 7^4 \equiv 1 ), then ( 7^{10} \equiv 1 )).

26. Sufficiency Logic for Factors/Multiples

27. On DS, never calculate unless necessary. Ask: "Does this statement guarantee the answer?"
28. When to use: Example: "Is ( n ) divisible by 15?" Statement: "( n ) is divisible by 30." → Sufficient (30 = 2 × 3 × 5).

Step-by-Step Strategy

For any integer/factor/multiple question, follow this process:

1. Identify the Core Question

2. Rewrite the question in your own words (e.g., "Does ( n ) have both 2 and 3 as factors?").

3. List Known Properties

4. Note given constraints (e.g., "( n ) is odd," "( n > 10 )") and relevant rules (e.g., "If ( n ) is divisible by 6, it must be divisible by 2 and 3").

5. Test Cases (If Needed)

6. For PS: Plug in numbers to eliminate choices (e.g., test ( n = 1, 2, 3 ) for "Which of the following must be true?").

7. For DS: Test extreme cases (e.g., smallest possible ( n ), largest possible ( n )) to check sufficiency.

8. Apply Prime Factorization (For Complex Problems)

9. Break numbers into primes to find GCF/LCM or test divisibility (e.g., "Is ( 12! + 1 ) divisible by 13?" → ( 12! ) includes 13 as a factor, so ( 12! + 1 \equiv 1 \mod 13 )).

10. Check for Traps

11. Common traps: Assuming ( n ) is positive, overlooking zero, or misapplying even/odd rules.

12. Verify the Answer

13. For PS: Ensure your answer matches the question’s requirements (e.g., "must be true" vs. "could be true").
14. For DS: Confirm both statements together don’t create a contradiction.

Fully Worked Example

Question:
If ( n ) is a positive integer, is ( n^2 - n ) divisible by 12? (1) ( n ) is divisible by 3.
(2) ( n ) is odd.

Step 1: Core Question
Does ( n^2 - n = n(n-1) ) have 12 as a factor? (i.e., divisible by 3 and 4).

Step 2: Known Properties
- ( n(n-1) ) is the product of two consecutive integers → always divisible by 2.
- For divisibility by 3: Either ( n ) or ( n-1 ) must be divisible by 3.
- For divisibility by 4: Either ( n ) or ( n-1 ) must be divisible by 4, or both must be even (since two evens multiply to a multiple of 4).

Step 3: Test Cases
- Statement (1): ( n ) is divisible by 3.
- Test ( n = 3 ): ( 3 \times 2 = 6 ) → Not divisible by 12. Insufficient.
- Test ( n = 6 ): ( 6 \times 5 = 30 ) → Not divisible by 12. Insufficient.
- Statement (2): ( n ) is odd.
- Test ( n = 1 ): ( 1 \times 0 = 0 ) → Divisible by 12 (0 is divisible by any number).
- Test ( n = 3 ): ( 3 \times 2 = 6 ) → Not divisible by 12. Insufficient.
- Statements (1) + (2): ( n ) is odd and divisible by 3.
- Let ( n = 3 ): ( 6 ) → No.
- Let ( n = 9 ): ( 9 \times 8 = 72 ) → Yes (72/12 = 6).
- Inconsistent results → Insufficient.

Step 4: Prime Factorization (Alternative Approach)
- ( n^2 - n = n(n-1) ). For divisibility by 12, need ( 2^2 \times 3 ).
- If ( n ) is odd, ( n-1 ) is even. For ( n(n-1) ) to be divisible by 4, ( n-1 ) must be divisible by 4 (since ( n ) is odd).
- Statement (1): Only guarantees divisibility by 3. Insufficient.
- Statement (2): Only guarantees ( n-1 ) is even. Insufficient.
- Combined: ( n ) is odd and divisible by 3. But ( n-1 ) may or may not be divisible by 4 (e.g., ( n = 3 ) vs. ( n = 9 )). Insufficient.

Answer: E (Neither statement alone nor together is sufficient).

Common Mistakes

1. Mistake: Assuming ( n ) is positive when the question allows zero or negatives.
2. Why it happens: GMAT often specifies "positive integer," but not always.

3. Correct approach: Check the question stem for constraints. If none, test ( n = 0 ) or ( n = -1 ).

4. Mistake: Misapplying even/odd rules (e.g., thinking "odd × odd = even").

5. Why it happens: Overlooking that odd × odd = odd.

6. Correct approach: Memorize the rules: Even × Any = Even; Odd × Odd = Odd.

7. Mistake: Forgetting that 1 is not a prime number.

8. Why it happens: 1 is often mistakenly included in prime factorizations.

9. Correct approach: Primes start at 2. 1 is neither prime nor composite.

10. Mistake: Overcomplicating DS questions by solving for exact values.

11. Why it happens: Students calculate ( n ) instead of testing sufficiency.

12. Correct approach: Ask: "Does this statement guarantee the answer?" If yes, it’s sufficient.

13. Mistake: Ignoring the "must be true" vs. "could be true" distinction.

14. Why it happens: Answering "could be true" for a "must be true" question.
15. Correct approach: For "must be true," test multiple cases to confirm consistency.

GMAT Traps & Timing

1. Trap: The "Zero" Trap
2. What it is: Questions involving divisibility often exclude zero, but not always.

3. How to avoid: If the question allows ( n = 0 ), test it (e.g., "Is ( n^2 - n ) divisible by 12?" → ( 0 ) is divisible by 12).

4. Trap: The "Negative" Trap

5. What it is: Assuming ( n ) is positive when it could be negative (e.g., "Is ( n^2 ) divisible by 4?" → ( n = -2 ) works).

6. How to avoid: Check the question stem for constraints.

7. Trap: The "LCM/GCF Swap"

8. What it is: Confusing LCM and GCF in word problems.
9. How to avoid: Remember: LCM is for "when will they coincide?"; GCF is for "what’s the largest shared piece?"

Time Budget:
- PS: 1–1.5 minutes per question.
- DS: 1.5–2 minutes per question (spend more time on statement analysis, less on calculation).

Quick Practice

1. Question: If ( n ) is a positive integer, is ( n^3 - n ) divisible by 4?
2. (1) ( n ) is odd.

3. (2) ( n ) is divisible by 3.
   Answer: A (Statement 1 alone is sufficient; ( n^3 - n = (n-1)n(n+1) ), and for odd ( n ), ( n-1 ) and ( n+1 ) are consecutive evens → divisible by 8, hence by 4).

4. Question: What is the greatest common factor of 48 and 72?
   Answer: 24 (Prime factors: ( 48 = 2^4 \times 3 ), ( 72 = 2^3 \times 3^2 ) → GCF = ( 2^3 \times 3 = 24 )).

Last-Minute Cram Sheet

1. Prime Factorization: Break numbers into primes to find GCF/LCM or test divisibility.
2. GCF: Product of the lowest power of common primes.
3. LCM: Product of the highest power of all primes.
4. Divisibility by 6: Must be divisible by 2 and 3.
5. Divisibility by 12: Must be divisible by 3 and 4.
6. Even × Any = Even; Odd × Odd = Odd.
7. Consecutive integers: Always include a multiple of 2, 3, etc.
8. Remainders: ( a = bq + r ), where ( 0 \leq r < b ).
9. DS Trap: "Sufficient" means the statement guarantees the answer, not just "could be true."
10. ⚠️ Zero is divisible by any integer (except zero).