🎯 GRE Quantitative Reasoning: Basic Arithmetic & Number Properties

📘 Introduction

Welcome to your GRE preparation journey! The Quantitative Reasoning section of the GRE tests your ability to understand, interpret, and analyze quantitative information. Don't worry—you don't need advanced mathematics. Instead, the GRE focuses on basic arithmetic, algebra, geometry, and data analysis, but presents them in clever, sometimes tricky ways.

This lesson covers the foundational arithmetic concepts and number properties that form the backbone of many GRE questions. Mastering these basics will give you the confidence and speed needed to tackle more complex problems.

💡 Pro Tip: The GRE isn't just testing what you know—it's testing how efficiently you can apply what you know under time pressure. We'll focus on strategies that save precious seconds!

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🧮 Core Concepts

1. Order of Operations (PEMDAS)

The order of operations determines the sequence in which you perform mathematical operations. Remember the acronym PEMDAS:

• Parentheses
• Exponents
• Multiplication and Division (left to right)
• Addition and Subtraction (left to right)

Example: 3 + 2 × (8 - 3)² ÷ 5

Step 1: Parentheses    → 3 + 2 × (5)² ÷ 5
Step 2: Exponents      → 3 + 2 × 25 ÷ 5
Step 3: Multiply/Divide → 3 + 50 ÷ 5
                       → 3 + 10
Step 4: Add/Subtract   → 13

⚠️ Common Mistake: Many students perform operations left to right without considering PEMDAS. Always handle multiplication and division before addition and subtraction!

2. Integer Properties

Integers are whole numbers that can be positive, negative, or zero: {..., -3, -2, -1, 0, 1, 2, 3, ...}

Key properties:

Even vs. Odd Numbers:

• Even: Divisible by 2 (e.g., -4, 0, 2, 18)
• Odd: Not divisible by 2 (e.g., -3, 1, 7, 25)

+----------------+--------+--------+
| Operation      | Result | Rule   |
+----------------+--------+--------+
| Even + Even    | Even   | 4 + 6  |
| Odd + Odd      | Even   | 3 + 5  |
| Even + Odd     | Odd    | 4 + 3  |
| Even × Even    | Even   | 4 × 6  |
| Even × Odd     | Even   | 4 × 3  |
| Odd × Odd      | Odd    | 3 × 5  |
+----------------+--------+--------+

🧠 Mnemonic: "Even times anything stays even; only odd times odd makes odd!"

Positive vs. Negative Numbers:

+----------------+----------+----------+
| Operation      | Result   | Example  |
+----------------+----------+----------+
| Pos × Pos      | Positive | 3 × 4    |
| Neg × Neg      | Positive | -3 × -4  |
| Pos × Neg      | Negative | 3 × -4   |
| Neg × Pos      | Negative | -3 × 4   |
+----------------+----------+----------+

💡 Quick Rule: Same signs multiply to positive; different signs multiply to negative.

3. Prime Numbers and Factors

Prime numbers are integers greater than 1 that have exactly two factors: 1 and themselves.

First 10 primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29

🔍 Key Insight: 2 is the only even prime number!

Factors are numbers that divide evenly into another number.

Example: Factors of 12 are: 1, 2, 3, 4, 6, 12

Prime Factorization breaks a number into its prime components:

         72
        /  \
       8  ×  9
      / \   / \
     2×4  3×3
      /\
     2×2

Prime factorization: 72 = 2³ × 3²

4. Divisibility Rules

These shortcuts help you quickly determine if one number divides evenly into another:

+--------+--------------------------------+------------+
| Number | Rule                           | Example    |
+--------+--------------------------------+------------+
|   2    | Last digit is even             | 348 ✓      |
|   3    | Sum of digits divisible by 3   | 123 (1+2+3=6) ✓ |
|   4    | Last 2 digits divisible by 4   | 316 (16÷4) ✓|
|   5    | Last digit is 0 or 5           | 425 ✓      |
|   6    | Divisible by both 2 AND 3      | 234 ✓      |
|   9    | Sum of digits divisible by 9   | 729 (7+2+9=18) ✓|
|   10   | Last digit is 0                | 570 ✓      |
+--------+--------------------------------+------------+

💡 Time-Saver: On test day, these rules can help you eliminate wrong answers in seconds!

5. Absolute Value

Absolute value |x| represents the distance from zero on a number line, always non-negative.

<----|----|----|----|----|----|----|----|---->
    -4   -3   -2   -1   0    1    2    3    4
     
     |-3| = 3  (distance from 0)
     |3| = 3   (distance from 0)

Key properties:

• |x| ≥ 0 for all x
• |x| = |-x|
• |x × y| = |x| × |y|

6. Fractions, Decimals, and Percentages

These three forms are interchangeable—being fluent in converting between them is crucial!

+------------+----------+-----------+
| Fraction   | Decimal  | Percent   |
+------------+----------+-----------+
| 1/2        | 0.5      | 50%       |
| 1/4        | 0.25     | 25%       |
| 3/4        | 0.75     | 75%       |
| 1/5        | 0.2      | 20%       |
| 1/10       | 0.1      | 10%       |
| 1/3        | 0.333... | 33.33%    |
| 2/3        | 0.666... | 66.67%    |
+------------+----------+-----------+

Converting Strategies:

• Fraction → Decimal: Divide numerator by denominator
• Decimal → Percent: Multiply by 100
• Percent → Decimal: Divide by 100
• Decimal → Fraction: Use place value (0.75 = 75/100 = 3/4)

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📚 Detailed Examples

Example 1: Complex Order of Operations

Problem: Calculate: 18 ÷ 3 + 2 × (7 - 4)² - 10

Solution:

Step 1: Parentheses first
18 ÷ 3 + 2 × (3)² - 10

Step 2: Exponents
18 ÷ 3 + 2 × 9 - 10

Step 3: Division and Multiplication (left to right)
6 + 18 - 10

Step 4: Addition and Subtraction (left to right)
24 - 10 = 14

Answer: 14

💡 Strategy: Write out each step clearly. Rushing through PEMDAS causes most arithmetic errors on the GRE.

Example 2: Even/Odd Number Properties

Problem: If x is an odd integer and y is an even integer, which of the following must be odd?

• x + y
• x × y
• x² + y
• x + y²
• 2x + y

Solution:

Let's test with x = 3 (odd) and y = 4 (even):

x + y = 3 + 4 = 7 (odd) ✓
x × y = 3 × 4 = 12 (even)
x² + y = 9 + 4 = 13 (odd) ✓
x + y² = 3 + 16 = 19 (odd) ✓
2x + y = 6 + 4 = 10 (even)

But we need to verify using rules, not just examples:

• x + y: odd + even = odd ✓
• x × y: odd × even = even
• x² + y: (odd)² + even = odd + even = odd ✓
• x + y²: odd + (even)² = odd + even = odd ✓
• 2x + y: 2(odd) + even = even + even = even

Multiple answers work! On the GRE, read carefully—if they ask "must be odd," all of the first, third, and fourth options are correct.

Example 3: Prime Factorization Application

Problem: What is the smallest positive integer that is divisible by both 12 and 18?

Solution:

Find the Least Common Multiple (LCM) using prime factorization:

12 = 2² × 3¹
18 = 2¹ × 3²

LCM = take highest power of each prime
    = 2² × 3²
    = 4 × 9
    = 36

Answer: 36

🔧 Try This: Verify by dividing: 36 ÷ 12 = 3 ✓ and 36 ÷ 18 = 2 ✓

Example 4: Percentage Problem

Problem: A jacket originally priced at $80 is on sale for 25% off. What is the sale price?

Solution:

Method 1 (Find discount, then subtract):

Discount = 25% of $80
         = 0.25 × 80
         = $20

Sale price = $80 - $20 = $60

Method 2 (Find remaining percentage directly):

If 25% off, you pay 75% of original
Sale price = 75% of $80
           = 0.75 × 80
           = $60

Answer: $60

💡 GRE Strategy: Method 2 is faster! If something is X% off, multiply by (100 - X)%.

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⚠️ Common Mistakes to Avoid

1. Ignoring Order of Operations

❌ Wrong: 5 + 3 × 2 = 8 × 2 = 16 ✅ Correct: 5 + 3 × 2 = 5 + 6 = 11

2. Mishandling Negative Numbers

❌ Wrong: -5² = 25 (thinking the negative sign is squared) ✅ Correct: -5² = -(5²) = -25 (only the 5 is squared) ✅ If you mean: (-5)² = 25 (use parentheses!)

3. Confusing "Divisible By" with "Divides Into"

"48 is divisible by 6" means 6 goes into 48 evenly (48 ÷ 6 = 8)

4. Assuming Zero is Positive

⚠️ Zero is neither positive nor negative—it's neutral!

5. Forgetting Special Cases

• 1 is NOT a prime number (it only has one factor: itself)
• 2 is the ONLY even prime number

6. Percentage Calculation Errors

"Increase by 50%" means multiply by 1.5, NOT add 50 "Decrease by 25%" means multiply by 0.75, NOT subtract 25

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🎯 Key Takeaways

✅ Master PEMDAS: Always follow the order of operations to avoid careless errors

✅ Know number properties: Even/odd, positive/negative rules save time on complex problems

✅ Memorize divisibility rules: Quick mental checks eliminate wrong answers instantly

✅ Prime factorization is your friend: Essential for LCM, GCF, and simplifying fractions

✅ Convert fluently: Move between fractions, decimals, and percentages with ease

✅ Use estimation: The GRE often allows you to eliminate wrong answers by approximating

✅ Practice mental math: Speed comes from doing calculations in your head when possible

🤔 Did You Know? The GRE provides an on-screen calculator, but using it for every calculation wastes time. Top scorers use it sparingly, relying on mental math for simple operations!

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📋 Quick Reference Card

+----------------------------------+------------------------+
| CONCEPT                          | KEY RULE               |
+----------------------------------+------------------------+
| Order of Operations              | PEMDAS                 |
| Even + Even                      | = Even                 |
| Odd × Odd                        | = Odd                  |
| Negative × Negative              | = Positive             |
| Only even prime                  | 2                      |
| 1 is prime?                      | NO                     |
| Divisibility by 3                | Sum of digits ÷ 3     |
| X% off means pay                 | (100-X)%               |
| |x|                              | Always ≥ 0             |
| LCM method                       | Highest prime powers   |
+----------------------------------+------------------------+

MEMORIZE THESE CONVERSIONS:
1/2 = 0.5 = 50%    |    1/4 = 0.25 = 25%
1/5 = 0.2 = 20%    |    1/3 ≈ 0.33 = 33%
3/4 = 0.75 = 75%   |    2/3 ≈ 0.67 = 67%

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📚 Further Study

1. Official GRE Quantitative Practice: https://www.ets.org/gre/test-takers/general-test/prepare.html
2. Khan Academy - GRE Math Prep: https://www.khanacademy.org/test-prep/gre
3. Manhattan Prep GRE Number Properties Guide: https://www.manhattanprep.com/gre/store/

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🎓 Congratulations! You've completed Lesson 1. These fundamental concepts appear in approximately 30-40% of GRE Quantitative questions. Master them, and you're well on your way to a competitive score!

💪 Next Steps: Practice these concepts with the questions below, then move on to Lesson 2: Algebra and Equations.