Integer Properties

Integer Properties Introduction and Breakdown

• The term "Integer Properties" refers to a broad spectrum of mathematical concepts involving whole numbers ($integers$) and the analysis of their composition.
• This subject is a core component of the GRE, appearing more frequently than Fractions, Decimals, Exponents, and Roots.
• Breakdown of Arithmetic, Algebra, and Number Properties:
  • Algebra: $35\%$ of questions.
  • Integer Properties: $25\%$ of questions.
  • Exponents & Roots: $20\%$ of questions.
  • Fractions or Decimals: $12.5\%$ of questions.
  • Functions: $7.5\%$ of questions.

Fundamental Concepts of Number Systems

• The Number Line:
  • A straight line where every point corresponds to a specific number and every number to a point.
  • Numbers left of zero are negative; numbers right of zero are positive.
  • Zero ($0$) is the origin point; it is neither positive nor negative.
  • Every point corresponds to a real number. Real numbers are all numbers that are not imaginary (all whole numbers, fractions, and decimals on the number line).
• Imaginary Numbers:
  • Defined as the square root of any negative number (e.g., $\sqrt{-4}$).
  • They are considered imaginary because no number produces a negative product when squared.
• Integers:
  • Whole numbers containing no decimal or fractional parts.
  • Examples of integers: Positive numbers ($1, 2, 3...$), negative numbers ($-1, -2, -3...$), and $0$.
  • Non-integers: Fractions (e.g., $\frac{1}{2}$, $\frac{5}{2}$), mixed numerals (e.g., $-2\frac{1}{2}$, $3\frac{4}{9}$), and decimals ($-2.1$, $3.4$).
• Squares and Perfect Squares:
  • The "square of an integer" is an integer resulting from squaring another integer (e.g., $36 = 6^2$, $81 = 9^2$, $169 = 13^2$, $225 = 15^2$).
  • A "perfect square" equals the square of a whole integer (e.g., $100 = 10^2$ is a perfect square; $50$ is not because $\sqrt{50} = 5\sqrt{2}$, which is not an integer).
• Properties of Zero ($0$):
  • It is an integer.
  • It is categorized as EVEN (like $2, 4, 6$), not odd.
  • It has three unique algebraic properties:
    1. $n \pm 0 = n$.
    2. $n \times 0 = 0$.
    3. Division by zero is impossible/undefined ($n \div 0 = \text{undefined}$).

Positive and Negative Numbers: Arithmetic and Logic

• Addition and Subtraction Rules:
  • Same signs: The result shares that sign (e.g., $7 + 8 = 15$, $-6 + (-2) = -8$, $-4 - 3 = -7$).
  • Different signs: Subtract the negative value from the positive (e.g., $-6 + 8 = 2$, $9 + (-6) = 3$, $4 - 7 = -3$).
  • Subtracting a negative: Equivalent to adding a positive number (e.g., $5 - (-6) = 5 + 6 = 11$, $-4 - (-3) = -4 + 3 = -1$).
• Multiplication and Division Rules:
  • Two negatives product/quotient is positive (e.g., $-6 \times (-6) = 36$, $-8 \div (-2) = 4$).
  • One positive and one negative product/quotient is negative (e.g., $5 \times (-6) = -30$, $6 \div (-3) = -2$).
  • Extended Rule: Multiplying an EVEN number of negative terms yields a positive product ($(-2) \times (-2) \times (-2) \times (-2) = 16$). Multiplying an ODD number of negative terms yields a negative product ($(-2) \times (-2) \times (-2) = -8$).
• Even Exponents and Variable Logic:
  • Any non-zero term raised to an EVEN exponent must be positive (e.g., $(-3)^2 = 9$, $(-2)^4 = 16$).
  • Simplifying inequalities: In expressions like $x^2y > 0$, remove even-powered terms to simplify ($y > 0$) because positive terms do not change the sign of the product.
  • Variable Caveat: Variables can represent any value (positive, negative, or zero) unless specified.

Absolute Value: Distance as Magnitude

• Definition:
  • Absolute value measures the distance between zero and a number on the number line.
  • It is denoted by two vertical bars: $|-4|$.
  • Because it represents distance, absolute value is NEVER negative. It is either positive or zero ($|0| = 0$).
  • Example: $|-4| = 4$ because $-4$ is "four steps from zero".

Number Line Dynamics and Interval Calculations

• Distance Between Points:
  • Method 1: Draw a number line and count steps between points.
  • Method 2: Take the absolute value of the difference between the two coordinates ($|x_1 - x_2|$). Example: Distance between $-4$ and $5$ is $|-4 - 5| = |-9| = 9$.
• Evenly Spaced Intervals:
  • An "interval" is the space between two tick marks.
  • The Number of Tick Marks is always ONE GREATER than the number of intervals ($\text{Ticks} = \text{Intervals} + 1$).
  • Length of an Interval Formula: $\frac{\text{Distance}}{\text{Number of Intervals}} = \text{Length of an Interval}$.
  • Example: Tree problem—$8$ trees planted at $15$-foot intervals. There are $7$ intervals, so total distance is $7 \times 15 = 105$ feet.
• Interpreting Diagrams:
  • "Drawn to Scale": You can assume tick marks are evenly spaced and approximate values precisely.
  • NOT "Drawn to Scale": You cannot assume even spacing or precise values. You only know the relative order (numbers increase rightward).
• Midpoints and Variables:
  • A midpoint divides a line segment into two equal lengths.
  • Strategy: Label individual intervals with variables (e.g., $x$ for segment $AB$, $y$ for segment $CD$) to create solvable equations.

Even and Odd Integers: Rules and Strategies

• Definitions:
  • Even: Integer divisible by $2$ (ends in $0, 2, 4, 6, 8$).
  • Odd: Integer not divisible by $2$ (ends in $1, 3, 5, 7, 9$).
  • Decimals and fractions are neither even nor odd.
  • Negative integers can be even or odd (e.g., $-2$ is even, $-3$ is odd).
• Calculation Outcomes:
  • Only TWO ways to generate an ODD number:
    1. Addition/Subtraction of one even and one odd ($E \pm O = O$).
    2. Multiplication of two odd numbers ($O \times O = O$).
  • All other combinations yield EVEN results:
    • $E \pm E = E$
    • $O \pm O = E$
    • $E \times E = E$
    • $E \times O = E$
  • Any string of numbers multiplied together will be EVEN if it contains at least one even number.
• Advanced Strategies:
  • Picking Numbers: Test two sets of integers. Set 1: Begins with an odd (e.g., $1, 2, 3$). Set 2: Begins with an even (e.g., $2, 3, 4$).
  • Factoring: Look for unfactored answer choices to reveal hidden properties ($mn - n$ factors to $n(m - 1)$).

Factors and Multiples: Definitions and Divisibility Rules

• **Factors