GMAT Review: Number Properties & Sets (Integers, Odd/Even, Prime, Divisibility)

GMAT – Number Properties & Sets (Integers, Odd/Even, Prime, Divisibility)

GMAT Study Guide – Number Properties & Sets
Focus: Integers, Odd/Even, Prime, Divisibility

What This Is

Number‑properties questions test your ability to reason about whole numbers—whether they’re odd or even, prime or composite, and how they relate through divisibility rules. They appear in both Problem‑Solving and Data‑Sufficiency items. A typical stem might read:

“If n is an integer such that n + 4 is even and n is a multiple of 3, which of the following could be the value of n?”

Mastering these concepts lets you eliminate answer choices quickly and avoid costly arithmetic errors.

Key Terms & Rules

• Even Integer: Any integer divisible by 2 (e.g., 8, –14).
• Odd Integer: Any integer not divisible by 2 (e.g., 7, –3).
• Prime Number: An integer > 1 whose only positive divisors are 1 and itself (e.g., 2, 5, 13).
• Composite Number: An integer > 1 that has at least one divisor other than 1 and itself (e.g., 4, 12).
• Divisibility Rule – 2, 5, 10: Look at the last digit (even → 0,2,4,6,8; 5‑multiple → 0 or 5; 10‑multiple → ends in 0).
• Divisibility Rule – 3 & 9: Sum of the digits; if the sum is divisible by 3 (or 9), the original number is.
• Divisibility Rule – 4 & 8: Last two digits (for 4) or last three digits (for 8) form a number divisible by the divisor.
• Parity Switch: Adding an even number preserves parity; adding an odd number flips parity.
• Prime‑Factor Shortcut: If a number is even, it cannot be prime except for 2.
• Data‑Sufficiency (DS) Answer Choices: A‑E format; remember that “Both statements together are sufficient” is C.
• Set Notation: {x | condition} denotes the set of all x satisfying the condition; useful for DS “Which of the following is true for all elements of the set?”

Step‑by‑Step Process Flow

1. Read the stem & isolate the property – Identify whether the question asks about even/odd, primality, or a divisibility condition.
2. Translate words into algebraic statements – e.g., “n is odd” → n = 2k + 1; “n is a multiple of 3” → n = 3m.
3. Apply the relevant rule – Use parity switches, digit‑sum rules, or factor‑based shortcuts to narrow possibilities.
4. Eliminate answer choices – Discard any choice that violates a rule; often 2–3 options drop instantly.
5. Confirm with a quick test – Plug the smallest viable integer into the original conditions; if it works, you’ve found the answer.

(For Data‑Sufficiency, after steps 1‑3, evaluate each statement separately, then together, and map the result to A‑E.)

Common Mistakes

• Mistake: Assuming “odd + odd = odd.”
  Correction: Odd + odd = even; only odd + even flips parity.

• Mistake: Forgetting that 2 is the only even prime.
  Correction: When a question asks “prime and even,” the answer must be 2; any other even number is automatically composite.

• Mistake: Applying the digit‑sum rule for 3 to a negative number without taking absolute value.
  Correction: Use the absolute value of the digits; the rule works on the magnitude, not the sign.

• Mistake: In DS, treating “n is a multiple of 6” as automatically implying “n is even.”
  Correction: While true, you must still verify the other condition (e.g., divisibility by 3) because the statement may be insufficient alone.

• Mistake: Over‑relying on “quick‑guess” arithmetic and ignoring the need for a proof in DS.
  Correction: Show a concrete example or counter‑example that satisfies the statements; the GMAT rewards logical justification over intuition.

Exam Insights

1. Parity is a frequent trap – Many PS items hide a parity flip in a multi‑step expression; look for “+ odd” or “– odd.”
2. Prime‑only questions often involve 2 or 3 – The GMAT loves the smallest primes because they limit the answer set dramatically.
3. Divisibility rules are tested in disguise – A DS stem may say “n leaves a remainder of 1 when divided by 3” → equivalent to “n + 2 is a multiple of 3.”
4. Set‑notation DS questions – Expect “For all x in the set {…}, is statement S true?” – you only need one counter‑example to prove insufficiency.

Quick Check Questions

1. Problem‑Solving:
   If an integer k is odd and k + 7 is a multiple of 4, which of the following could be the value of k?
   Answer: B (5) – 5 + 7 = 12, which is divisible by 4; other choices fail the parity or divisibility test.

2. Data‑Sufficiency:
   Question: Is n a prime number?
   
   1) n is odd and greater than 2.
   
   2) n has no divisors other than 1 and n that are less than 10.
   Answer: C – Statement 1 alone is insufficient (odd numbers can be composite); Statement 2 alone is insufficient (e.g., 121 = 11²); together they guarantee n < 100 and prime, so sufficient.

3. Quantitative Comparison:
   Column A: The smallest positive integer that is a multiple of 6 but not a multiple of 4.
   Column B: 12
   Answer: A > B – The smallest such integer is 6 (multiple of 6, not of 4), which is less than 12, so Column A (6) < Column B (12). Correct answer: B.

Last‑Minute Cram Sheet (10 One‑Liners)

1. ⚠️ Parity Rule: Even + Even = Even; Odd + Odd = Even; Odd + Even = Odd.
2. Only even prime is 2 – any other even integer is composite.
3. Divisibility by 3/9: Sum of digits divisible by 3 (or 9).
4. Divisibility by 4: Last two digits form a number divisible by 4.
5. Divisibility by 5: Ends in 0 or 5; by 10 ends in 0.
6. Multiple of 6 → must be even and divisible by 3.
7. Prime‑factor shortcut: If a number is > 2 and even, it’s automatically non‑prime.
8. DS Answer‑Choice Map: A = (1) only, B = (2) only, C = both, D = either, E = insufficient.
9. Set notation tip: “For all x in {…}” → a single counter‑example disproves the statement.
10. ⚠️ Never assume extra info – the GMAT never gives you hidden constraints; work only with what’s explicitly stated.