Mastering GMAT Number Properties: A Postgraduate Revision Guide

GMAT Number Properties: The Core Concepts

The GMAT examiner tests your foundational understanding of integers, their classifications (e.g., prime, composite, even, odd), and how they interact through operations like division, multiplication, and exponentiation. Specifically, they assess your ability to apply these principles to determine divisibility, find remainders, and analyze the properties of numbers under various constraints.

The Method: A Step-by-Step Approach

When faced with a GMAT Number Properties question, follow these steps to ensure a systematic and accurate solution:

1. Deconstruct the Question: Identify all given conditions, constraints, and the specific quantity or property you need to determine. Pay close attention to keywords like "integer," "prime," "remainder," "divisible by," and "positive/negative."

2. Translate to Mathematical Notation: Convert verbal descriptions into algebraic expressions or number theory statements. For example, "when \(N\) is divided by \(D\), the remainder is \(R\)" translates to \(N = QD + R\), where \(Q\) is the quotient and \(0 \le R < D\).

3. Apply Relevant Number Properties: Based on the translated information, recall and apply the pertinent number properties. This might involve:

   • Prime Factorization: Breaking numbers down into their prime components to analyze divisibility.
   • Divisibility Rules: Using shortcuts for common divisors (e.g., by 3, 4, 5, 9).
   • Remainder Theorem: Understanding the relationship \(N = QD + R\).
   • Even/Odd Properties: Analyzing how sums, differences, and products of even/odd integers behave.
   • Consecutive Integers: Representing them algebraically (e.g., \(n, n+1, n+2\)).

4. Test Cases or Algebraic Manipulation:

   • Test Cases (for Data Sufficiency): If the question is Data Sufficiency, pick small, valid numbers that satisfy the given conditions to test the sufficiency of each statement. Ensure you test edge cases (e.g., small primes, negative numbers if allowed, zero if allowed).
   • Algebraic Manipulation (for Problem Solving): If it's a Problem Solving question, use algebraic techniques to simplify expressions, solve equations, or prove a property.

5. Verify and Conclude: Double-check your calculations and ensure your answer directly addresses the question asked. For Data Sufficiency, confirm that your chosen numbers definitively prove or disprove sufficiency.

Fully Worked Example

Question: When a positive integer \(X\) is divided by 7, the remainder is 3. When \(X\) is divided by 5, the remainder is 2. What is the smallest positive integer \(X\) that satisfies these conditions?

Step 1: Deconstruct the Question
We are looking for the smallest positive integer \(X\).
Condition 1: \(X\) divided by 7 leaves a remainder of 3.
Condition 2: \(X\) divided by 5 leaves a remainder of 2.

Step 2: Translate to Mathematical Notation
Condition 1: \(X = 7Q_1 + 3\) for some integer \(Q_1 \ge 0\). This implies \(X \equiv 3 \pmod{7}\).
Condition 2: \(X = 5Q_2 + 2\) for some integer \(Q_2 \ge 0\). This implies \(X \equiv 2 \pmod{5}\).

Step 3: Apply Relevant Number Properties
We are dealing with simultaneous congruences. We need to find a number that fits both remainder conditions.

Step 4: Test Cases or Algebraic Manipulation
Let's list possible values for \(X\) based on the first condition:
\(X = 7(0) + 3 = 3\)
\(X = 7(1) + 3 = 10\)
\(X = 7(2) + 3 = 17\)
\(X = 7(3) + 3 = 24\)
\(X = 7(4) + 3 = 31\)
\(X = 7(5) + 3 = 38\)
\(X = 7(6) + 3 = 45\)
...

Now, let's check which of these satisfy the second condition (\(X\) divided by 5 leaves a remainder of 2):
* \(3 \div 5\), remainder is 3. (No)
* \(10 \div 5\), remainder is 0. (No)
* \(17 \div 5\), remainder is 2. (Yes!)

So, \(X=17\) is a candidate. Since we listed numbers in increasing order, 17 is the smallest positive integer that satisfies both conditions.

Alternatively, using algebraic manipulation:
From \(X = 7Q_1 + 3\), we substitute this into the second congruence:
\(7Q_1 + 3 \equiv 2 \pmod{5}\)
\(2Q_1 + 3 \equiv 2 \pmod{5}\) (since \(7 \equiv 2 \pmod{5}\))
\(2Q_1 \equiv -1 \pmod{5}\)
\(2Q_1 \equiv 4 \pmod{5}\)
Since 2 and 5 are coprime, we can divide by 2:
\(Q_1 \equiv 2 \pmod{5}\)
So, \(Q_1\) can be 2, 7, 12, etc.
The smallest positive \(Q_1\) is 2.
Substitute \(Q_1 = 2\) back into \(X = 7Q_1 + 3\):
\(X = 7(2) + 3 = 14 + 3 = 17\).

Step 5: Verify and Conclude
For \(X=17\):
\(17 \div 7 = 2\) with a remainder of \(3\). (Correct)
\(17 \div 5 = 3\) with a remainder of \(2\). (Correct)
The smallest positive integer \(X\) is 17.

Three Mistakes That Lose Marks

1. Ignoring "Positive Integer" or "Integer": Assuming numbers are positive when they could be negative, or assuming they are integers when they could be fractions, leads to incorrect conclusions, especially in Data Sufficiency. Always check the domain of the variables.

2. Misapplying Remainder Definition: The remainder \(R\) when \(N\) is divided by \(D\) must satisfy \(0 \le R < D\). Forgetting this can lead to errors, particularly with negative numbers or when manipulating remainder equations. For instance, a remainder of -2 when dividing by 5 is equivalent to a remainder of 3.

3. Insufficient Test Cases (Data Sufficiency): When testing statements in Data Sufficiency, students often pick only one set of numbers that satisfy the condition and conclude sufficiency. You must try to find a counterexample (another set of numbers that satisfies the condition but gives a different answer to the question) to truly prove insufficiency.

30-Second Recap

GMAT Number Properties requires precise application of definitions for primes, divisibility, and remainders. Translate questions into mathematical statements, apply relevant properties (prime factorization, remainder theorem), and systematically test cases or use algebraic manipulation. Crucially, always respect integer constraints, the strict definition of remainders, and thoroughly test multiple scenarios in Data Sufficiency to avoid common pitfalls.