Quant
Master the fundamental distinction between even and odd numbers to unlock powerful shortcuts and strategic advantages on the GMAT Focus Edition quantitative section.
The Power of Parity in Quantitative Reasoning
Beyond their numerical value, integers possess an inherent characteristic known as parity 3 whether they are even or odd. This seemingly simple distinction is a powerful tool in quantitative reasoning, particularly on the GMAT Focus Edition.
Understanding how even and odd numbers behave under various arithmetic operations can unlock shortcuts, simplify complex problems, and allow you to eliminate incorrect answer choices efficiently. This lesson delves deeply into definitions, operational rules, and strategic implications.
Defining Even and Odd Numbers
The Core Distinction
The classification of an integer as even or odd is based on its divisibility by 2. This fundamental property determines how numbers interact in arithmetic operations.
Formal Definitions
Even numbers are divisible by 2 with no remainder (form: 2n). Odd numbers have a remainder of 1 when divided by 2 (form: 2n + 1).
Even Numbers: Characteristics and Properties
Formal Definition
An integer is even if it is divisible by 2 with no remainder. This means an even number can be expressed in the form 2n, where n is any integer.
Key Characteristics
Always end in digits 0, 2, 4, 6, or 8
Can be positive (2, 4, 6, ...), negative (-2,-4,-6, ...), or zero (0)
Divisible by 2 without remainder
Special Case: Zero
Zero is an even number because it can be expressed as 2 × 0 (where n=0 is an integer), and it is divisible by 2 with a remainder of 0. This is a common GMAT trap!
Odd Numbers: Characteristics and Properties
Formal Definition
An integer is odd if it is notdivisible by 2 (i.e., it has a remainder of 1 when divided by 2). This means an odd number can be expressed in the form 2n + 1 or 2n - 1, where n is any integer.
Key Characteristics
Always end in digits 1, 3, 5, 7, or 9
Can be positive (1, 3, 5, ...) or negative (-1,-3,-5, ...)
Leave remainder of 1 when divided by 2
Parity Rules: Addition and Subtraction
Understanding how even and odd numbers interact under addition and subtraction is fundamental. The rules for both operations are identical.
Even + Even = Even
Example: 4 + 6 = 10
Algebraic: 2n + 2m = 2(n+m)
Odd + Odd = Even
Example: 3 + 5 = 8
Algebraic: (2n+1) + (2m+1) = 2(n+m+1)
Even + Odd = Odd
Example: 4 + 3 = 7
Algebraic: 2n + (2m+1) = 2(n+m) + 1
Key Insight: Same Parity (Even + Even, Odd + Odd) = Even | Different Parity (Even + Odd) = Odd www.goalisb.com | Shruti P | ISB, IIM, MBA Expert | contact@goalisb.com | https://www.youtube.com/@Goalisb
Parity Rules: Multiplication
Even × Even = Even
Example: 4 × 6 = 24
Algebraic: (2n) × (2m) = 4nm = 2(2nm)
Odd × Odd = Odd
Example: 3 × 5 = 15
Algebraic: (2n+1) × (2m+1) = 2(2nm + n + m) + 1
Even × Odd = Even
Example: 4 × 3 = 12
Algebraic: (2n) × (2m+1) = 2(2nm + n)
Critical Rule: If any factor is Even, the product is Even. The product is Odd ONLY if allfactors are Odd.
Parity Rules: Division
Parity rules are not as straightforward for division because the result may not be an integer. Rules apply only when the result is an integer.
Even ÷ Even
Can be Even, Odd, or not an integer 10 ÷ 2 = 5 (Odd) 12 ÷ 2 = 6 (Even) 10 ÷ 4 = 2.5 (Not integer) Odd ÷ Odd
be Odd or not an integer
÷ 7 (Not integer) Even ÷ Odd Can be Even or not an integer
÷ 5 = 2 (Even)
÷ 3 = 4 (Even)
÷ 3 (Not integer) Odd ÷ Even Can NEVER be an integer
If an odd number were divisible by an even number, it would imply the odd number is divisible by 2, contradicting its definition.
Algebraic Representation of Parity
Standard Forms
Let n be any integer:
Even integer: 2n
Odd integer: 2n + 1 (or 2n - 1)
Example Proof: Odd + Odd = Even
Let first odd number be 2n + 11.
Let second odd number be 2m + 12.
Sum: (2n + 1) + (2m + 1) = 2n + 2m + 23.
Factor out 2: 2(n + m + 1)4.
Since n and m are integers, n + m + 1 is also an in teger5.
The sum is in form 2 × (integer), which defines an even number. Q.E.D.6.
Using algebraic forms like 2n and 2n+1 is invaluable for proving parity properties or solving problems where specific numbers aren't given.
Why Parity Matters for GMAT Success
Quick Elimination
Knowing parity rules allows you to eliminate answer choices instantly. If a problem asks for an even result and you get an odd one (or vice versa), you can rule out options immediately.
Data Sufficiency
Parity questions are very common in Data Sufficiency. Understanding the rules helps you determine if a given statement provides enough information about the parity of a variable or expression.
Number Properties
Many questions directly test your understanding of even/odd properties, especially in combination with other concepts like divisibility or remainders.
Strategic Guessing
If you're stuck, applying parity rules can sometimes increase your odds of guessing correctly by eliminating impossible answer choices.
Common Pitfalls to Avoid
1
Zero's Parity
Forgetting that 0 is an even number is a very common mistake. Always remember 0 = 2 × 0.
2
Negative Numbers
Applying parity rules to negative numbers incorrectly. The rules apply identically to positive and negative integers.-4 is even; -3 is odd.
4
Fractional Results
Parity applies only to integers. 3 ÷ 2 = 1.5 is neither even nor odd.
Interactive Check Your Understanding:
Is -100 an even or odd number?1.
If X is an odd integer, and Y is an even integer, w hat is the parity of XY?2.
If A and B are integers, and A × B is odd, what can you conclude about the parity of A and B?3.
Can 17 ÷ Z be an integer if Z is an even number? Ex plain.4.
3
Division Assumptions
Do not assume Even ÷ Even or Even ÷ Odd will always result in an integer with specific parity. Always check for integer results. Odd ÷ Even never results in an integer.
5
Misapplication of Rules
Double-check your understanding: "same parity yields even for +/-" and "any even factor yields even for ×."
Practice Questions and Solutions
Practice Questions:
Which of the following expressions will result in a n odd integer?1.
a) (Even + Even) × Odd
b) (Odd - Even) + Even
c) (Odd × Odd) + Even
d) (Even × Odd) - Odd
If P is an even integer and Q is an odd integer, wh ich must be an even integer?2.
a) P + Q + 1
b) P × Q - 3
c) P / 2 + Q
d) (P - Q)²
The sum of three consecutive integers is always:3.
a) Even
b) Odd
c) A multiple of 3
d) Prime
If x and y are integers, and x + y is odd, which mu st be true?4.
a) x is even and y is even
b) x is odd and y is odd
c) x and y have different parities
d) x × y is odd
5.
Consider (N + 1) × N × (N - 1). If N is an integer, what can be said about the parity of this product?
a) It is always odd
b) It is always even
c) It can be either even or odd
d) It is always a multiple of 4
Detailed Solutions:
Expression Analysis
Options b, c, and d all result in odd integers. Answer: b) (Odd - Even) + Even = Odd + Even = Odd
Even/Odd Combination
P + Q + 1: Even + Odd + 1 = Odd + 1 = Even. Answer: a) P + Q + 1
Consecutive Integers
Sum = n + (n+1) + (n+2) = 3n + 3 = 3(n+1). Always divisible by 3. Answer: c) A multiple of 3
Parity Difference
Sum is odd only when parities are different. Answer: c) x and y have different parities
Consecutive Product
Among three consecutive integers, at least one is even, making the product always even. Answer: b) It is always even
Comprehensive GMAT Resources
GMAT Focus Edition
Courses
GMAT Focus Edition Online
Course
GMAT Focus Edition Course
Quant
GMAT Quantitative Skills
Course
Additional Resources
GMAT Exam Dates
GMAT Syllabus
GMAT Preparation
Global GMAT Trends 200532024
Test Comparisons
Executive Assessment vs GMAT vs GRE
GMAT vs CAT
GMAT vs GRE
Specialized Topics
Decimal Mastery
Number Properties
Prime Numbers
Properties of Exponents
Data Insights Mastery
Mastering GMAT Data Insights
Graphics Interpretation
Two-Part Analysis
Table Analysis
MBA Programs
Top Deferred MBA Programs
IIMA BPGP
Talk To An Expert