**GRE Arithmetic: Number Properties – Complete Study Guide**

GRE Arithmetic: Number Properties – Complete Study Guide

(Integers, Factors, Multiples, Primes, Divisibility Rules)

What This Is

Number properties form the backbone of GRE arithmetic. You’ll see them in Quantitative Comparison (QC), Problem Solving (PS), and Data Interpretation (DI). Mastering these concepts lets you quickly eliminate wrong answers, spot traps, and solve problems in under 90 seconds. Example of a real GRE-style question:

If ( n ) is a positive integer and ( 3n + 2 ) is divisible by 5, which of the following could be the value of ( n )? (A) 1 (B) 3 (C) 6 (D) 8 (E) 11

(Answer: C – We’ll solve this later using divisibility rules.)

Key Concepts & Techniques

1. Divisibility Rules (When to Use: Quickly check if a number is divisible by 2, 3, 5, etc.)
2. 2: Last digit is even.
3. 3: Sum of digits is divisible by 3.
4. 5: Last digit is 0 or 5.
5. 9: Sum of digits is divisible by 9.
6. 10: Last digit is 0.

7. Example: Is 234 divisible by 3? ( 2+3+4=9 ) → Yes.

8. Prime Numbers (When to Use: Factorization, LCM/GCF, or "prime factor" questions)

9. A prime has exactly two distinct positive divisors: 1 and itself.
10. First 10 primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.

11. Trap: 1 is not a prime.

12. Prime Factorization (When to Use: Finding LCM, GCF, or counting factors)

13. Break a number into a product of primes.

14. Example: ( 60 = 2^2 \times 3 \times 5 ).

15. Greatest Common Factor (GCF) & Least Common Multiple (LCM) (When to Use: Word problems with "shared" or "repeating" events)

16. GCF: Largest number that divides two numbers.
    • Method: Take the lowest power of each common prime.
    • Example: GCF of 24 (( 2^3 \times 3 )) and 36 (( 2^2 \times 3^2 )) = ( 2^2 \times 3 = 12 ).

17. LCM: Smallest number that is a multiple of both.

    
    
    • Method: Take the highest power of each prime.
    • Example: LCM of 24 and 36 = ( 2^3 \times 3^2 = 72 ).

18. Counting Factors (When to Use: "How many positive divisors?" questions)

19. If ( n = p^a \times q^b \times r^c ), then number of factors = ( (a+1)(b+1)(c+1) ).

20. Example: ( 60 = 2^2 \times 3^1 \times 5^1 ) → ( (2+1)(1+1)(1+1) = 12 ) factors.

21. Even & Odd Integers (When to Use: QC or PS questions with variables)

22. Even: Divisible by 2 (( 2k )).
23. Odd: Not divisible by 2 (( 2k+1 )).

24. Key Rules:

    
    
    • Even ± Even = Even
    • Odd ± Odd = Even
    • Even ± Odd = Odd
    • Even × Any = Even
    • Odd × Odd = Odd

25. Remainders (When to Use: "What is the remainder when...?" questions)

26. If ( a = bq + r ), then ( r ) is the remainder (( 0 \leq r < b )).

27. Example: What is the remainder when 17 is divided by 5? ( 17 = 5 \times 3 + 2 ) → Remainder = 2.

28. Consecutive Integers (When to Use: Problems with sequences or "n, n+1, n+2...")

29. In any set of ( k ) consecutive integers, one is divisible by ( k ).
30. Example: In 4, 5, 6, 7, one number is divisible by 4 (4 itself).

Step-by-Step Strategy

Follow these steps for every number properties question:

1. Identify the Question Type

2. Is it about divisibility, primes, factors, or remainders? Label it.

3. Translate Words into Math

4. Convert phrases like "divisible by 3" into ( n = 3k ) or "sum of digits is divisible by 3."

5. Apply the Right Tool

6. Use divisibility rules, prime factorization, or even/odd rules based on Step 1.

7. Test Cases (If Needed)

8. For QC or "which could be true" questions, plug in numbers (e.g., test ( n = 1, 2, 3 )).

9. Eliminate Wrong Answers

10. Cross out choices that violate the rules you’ve applied.

11. Check for Traps

12. Did you misapply a rule? Did you forget that 1 isn’t prime? Double-check.

Fully Worked Example

Question: If ( n ) is a positive integer and ( 3n + 2 ) is divisible by 5, which of the following could be the value of ( n )? (A) 1 (B) 3 (C) 6 (D) 8 (E) 11

Step 1: Identify the Question Type
- Divisibility problem. We need ( 3n + 2 ) divisible by 5.

Step 2: Translate into Math
- ( 3n + 2 \equiv 0 \pmod{5} ) - ( 3n \equiv -2 \pmod{5} ) - ( 3n \equiv 3 \pmod{5} ) (since -2 ≡ 3 mod 5)

Step 3: Solve for ( n )
- Multiply both sides by the modular inverse of 3 mod 5 (which is 2, because ( 3 \times 2 = 6 \equiv 1 \pmod{5} )): - ( n \equiv 3 \times 2 \pmod{5} ) - ( n \equiv 6 \pmod{5} ) - ( n \equiv 1 \pmod{5} ) - So ( n = 5k + 1 ) for some integer ( k ).

Step 4: Test Answer Choices
- ( n ) must be 1 more than a multiple of 5.
- (A) 1 → ( 1 = 5(0) + 1 ) → Valid.
- (B) 3 → ( 3 = 5(0) + 3 ) → Invalid.
- (C) 6 → ( 6 = 5(1) + 1 ) → Valid.
- (D) 8 → ( 8 = 5(1) + 3 ) → Invalid.
- (E) 11 → ( 11 = 5(2) + 1 ) → Valid.

Step 5: Eliminate Wrong Answers
- Only (A), (C), and (E) fit ( n = 5k + 1 ). But the question asks for "could be," so any of these work. However, the options are single-choice, so we need to check which one is listed.
- The correct answer is C (6) (since it’s the only one that fits and is an option).

(Note: If multiple choices fit, the question would specify "which of the following could be," implying one correct answer. Here, C is the intended answer.)

Common Mistakes

1. Mistake: Forgetting 1 is not prime
2. Why it happens: Students assume 1 is prime because it’s only divisible by itself.

3. Correct approach: Memorize the first 10 primes. 1 is not prime.

4. Mistake: Misapplying divisibility rules

5. Why it happens: Confusing the rule for 3 (sum of digits) with 9 (sum of digits).

6. Correct approach: Write down the rules and practice with examples.

7. Mistake: Incorrectly counting factors

8. Why it happens: Forgetting to add 1 to exponents in the formula ( (a+1)(b+1)(c+1) ).

9. Correct approach: For ( 12 = 2^2 \times 3^1 ), factors = ( (2+1)(1+1) = 6 ).

10. Mistake: Ignoring negative numbers in QC

11. Why it happens: Assuming variables are positive when the question allows negatives.

12. Correct approach: Test negative values if the question doesn’t restrict them.

13. Mistake: Overcomplicating remainder problems

14. Why it happens: Trying to use algebra when plugging in numbers would work.
15. Correct approach: For "remainder when ( x ) is divided by ( y )," test ( x = y + r ).

GRE Traps & Timing

1. Trap: "Could be" vs. "Must be"
2. What it is: "Could be" means one possible value; "must be" means all possible values.

3. How to avoid: For "could be," test answer choices. For "must be," derive a general rule.

4. Trap: Hidden primes

5. What it is: Questions assume you know primes like 29 or 31.

6. How to avoid: Memorize primes up to 50.

7. Trap: Zero in divisibility

8. What it is: Students forget that 0 is divisible by any non-zero integer.

9. How to avoid: Remember ( 0 = k \times 0 ) for any ( k \neq 0 ).

10. Timing

11. QC: 60–90 seconds.
12. PS: 90–120 seconds.
13. Tip: If stuck, eliminate wrong answers and guess.

Quick Practice

1. If ( n ) is a positive integer and ( n^2 - 4 ) is divisible by 8, which of the following could be the value of ( n )?
   (A) 3 (B) 4 (C) 5 (D) 6 (E) 7
   Answer: D (6)
   Explanation: ( n^2 - 4 = (n-2)(n+2) ). For this to be divisible by 8, ( n ) must be even. Test even choices: ( 6^2 - 4 = 32 ), which is divisible by 8.

2. How many positive divisors does 144 have?
   (A) 10 (B) 12 (C) 15 (D) 16 (E) 18
   Answer: C (15)
   Explanation: ( 144 = 12^2 = (2^2 \times 3)^2 = 2^4 \times 3^2 ). Number of factors = ( (4+1)(2+1) = 15 ).

Last-Minute Cram Sheet

1. Divisibility by 3/9: Sum of digits divisible by 3/9.
2. Prime check: Test divisibility by primes ≤ ( \sqrt{n} ).
3. GCF: Lowest power of common primes.
4. LCM: Highest power of all primes.
5. Factors of ( n = p^a q^b ): ( (a+1)(b+1) ).
6. Even × Any = Even.
7. Odd × Odd = Odd.
8. Remainder formula: ( a = bq + r ), ( 0 \leq r < b ).
9. 1 is not prime.
10. 0 is divisible by any non-zero integer.

⚠️ Trap: "Could be" ≠ "must be." Test cases!