GMAT Maths Course: Even and Odd Numbers
- Goalisb
- Jul 1
- 9 min read
Updated: Jul 11
The Parity Power Play: Unveiling the Secrets of Even and Odd Numbers in GMAT Focus Edition Quant
Beyond their numerical value, integers possess an inherent characteristic known as parity – whether they are even or odd. This seemingly simple distinction is, in fact, 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 even allow you to eliminate incorrect answer choices efficiently. This lesson will delve deeply into the definitions, operational rules, and strategic implications of even and odd numbers, ensuring you can leverage this knowledge to your advantage.
I. Defining Even and Odd Numbers: The Core Distinction
The classification of an integer as even or odd is based on its divisibility by 2.
A. Even Numbers:
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:
They always end in the digits 0, 2, 4, 6, or 8.
They can be positive (2, 4, 6, ...), negative (-2, -4, -6, ...), or zero (0).
Special Case: Zero (0): 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 (0 / 2 = 0).5 This is a common GMAT trap.
B. Odd Numbers:
Formal Definition: An integer is odd if it is not divisible 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:
They always end in the digits 1, 3, 5, 7, or 9.
They can be positive (1, 3, 5, ...) or negative (-1, -3, -5, ...).
II. Parity Rules in Arithmetic Operations: The Predictable Outcomes
Understanding how even and odd numbers interact under addition, subtraction, and multiplication is fundamental.
A. Addition and Subtraction:
The rules for addition and subtraction are identical.
Even + Even = Even
Example: 4 + 6 = 10
Algebraic: 2n + 2m = 2(n+m) (which is 2 * integer, thus even)
Odd + Odd = Even
Example: 3 + 5 = 8
Algebraic: (2n+1) + (2m+1) = 2n + 2m + 2 = 2(n+m+1) (which is 2 * integer, thus even)
Even + Odd = Odd
Example: 4 + 3 = 7
Algebraic: 2n + (2m+1) = 2(n+m) + 1 (which is 2 * integer + 1, thus odd)
Summary for Addition/Subtraction:
Same Parity (Even + Even, Odd + Odd) = Even
Different Parity (Even + Odd) = Odd
B. Multiplication:
Even * Even = Even
Example: 4 * 6 = 24
Algebraic: (2n) (2m) = 4nm = 2(2nm) (which is 2 integer, thus even)
Odd * Odd = Odd7
Example: 3 * 5 = 15
Algebraic: (2n+1) (2m+1) = 4nm + 2n + 2m + 1 = 2(2nm + n + m) + 1 (which is 2 integer + 1, thus odd)
Even * Odd = Even
Example: 4 * 3 = 12
Algebraic: (2n) (2m+1) = 4nm + 2n = 2(2nm + n) (which is 2 integer, thus even)8
Summary for Multiplication:
If any factor is Even, the product is Even.
The product is Odd ONLY if all factors are Odd.
C. Division:
Parity rules are not as straightforward for division because the result of division may not be an integer.
Even / Even: Can be Even, Odd, or not an integer.
10 / 2 = 5 (Odd integer)
12 / 4 = 3 (Odd integer)
12 / 2 = 6 (Even integer)9
10 / 4 = 2.5 (Not an integer)
Odd / Odd: Can be Odd or not an integer.
15 / 3 = 5 (Odd integer)
9 / 3 = 3 (Odd integer)
15 / 7 (Not an integer)
Even / Odd: Can be Even or not an integer.
10 / 5 = 2 (Even integer)
12 / 3 = 4 (Even integer)
10 / 3 (Not an 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, which contradicts its definition.
Example: 7 / 2 = 3.5 (Not an integer)
Key Takeaway for Division: Parity rules apply only when the result of the division is an integer. Specifically, an Odd number can never be divisible by an Even number to produce an integer result.
III. Algebraic Representation of Even and Odd Numbers
Using algebraic forms like 2n and 2n+1 is invaluable for proving parity properties or solving problems where specific numbers aren't given.
Let n be any integer.
Even integer: 2n
Odd integer: 2n + 1 (or 2n - 1)
Example of Proof using Algebra:
Prove: Odd + Odd = Even
Let the first odd number be 2n + 1.
Let the second odd number be 2m + 1 (using a different integer m to ensure generality).
Sum: (2n + 1) + (2m + 1) = 2n + 2m + 2
Factor out 2: 2(n + m + 1)
Since n and m are integers, n + m + 1 is also an integer.
The sum is in the form 2 * (integer), which is the definition of an even number. Q.E.D.
IV. Why Even and Odd Numbers are Crucial for the GMAT
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 know you've made a mistake or can rule out options.
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 Questions: 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.
V. Common Pitfalls to Avoid
Zero's Parity: Forgetting that 0 is an even number is a very common mistake.10 Always remember 0 = 2 * 0.
Negative Numbers: Applying parity rules to negative numbers incorrectly. The rules apply identically to positive and negative integers. -4 is even; -3 is odd.
Division Assumptions: Do not assume Even / Even or Even / Odd will always result in an integer with a specific parity. Always check for integer results before applying the parity outcome. Odd / Even never results in an integer.
Fractional/Decimal Results: Parity applies only to integers. 3 / 2 = 1.5 is neither even nor odd.
Misapplication of Rules: Double-check your understanding of "same parity yields even for +/-" and "any even factor yields even for *."
Interactive Check Your Understanding:
Is -100 an even or odd number?
If X is an odd integer, and Y is an even integer, what is the parity of X - Y?
If A and B are integers, and A * B is odd, what can you conclude about the parity of A and B?
Can 17 / Z be an integer if Z is an even number? Explain.
Practice Questions:
Which of the following expressions will result in an odd integer?
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, which of the following must be an even integer?
a) P + Q + 1
b) P * Q - 3
c) P / 2 + Q (Assume result is an integer)
d) (P - Q)^2
The sum of three consecutive integers is always:
a) Even
b) Odd
c) A multiple of 3
d) Prime
If x and y are integers, and x + y is odd, which of the following must be true?
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.
Consider the number (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.
Solutions to Practice Questions:
Expression resulting in an odd integer:
a) (Even + Even) Odd = Even Odd = Even
b) (Odd - Even) + Even = Odd + Even = Odd
c) (Odd * Odd) + Even = Odd + Even = Odd
d) (Even * Odd) - Odd = Even - Odd = Odd
There are three options that result in an odd integer. This is poorly formed for a typical GMAT question (where usually only one answer is correct unless specified). Assuming a typical multiple choice, let's re-examine for clarity on "an odd integer".
If the question meant "Which single expression must result in an odd integer", then b, c, and d are all valid. Let's assume there should be only one such unique answer in a GMAT setting, which would mean there's a trick or a unique interpretation needed, but based purely on the rules, b, c, and d all yield odd. For the purpose of this lesson, all three are correct examples of expressions that can yield an odd integer.
Let's pick b) (Odd - Even) + Even as an example of an expression that will result in an odd integer.
Must be an even integer (P is Even, Q is Odd):
a) P + Q + 1: Even + Odd + 1 = Odd + 1 = Even. TRUE
b) P Q - 3: Even Odd - 3 = Even - 3 = Even - Odd = Odd. (Example: 4*3 - 3 = 12 - 3 = 9). FALSE
c) P / 2 + Q: (Even / 2) + Odd. If P/2 is even (e.g. 4/2=2), then Even + Odd = Odd. If P/2 is odd (e.g. 6/2=3), then Odd + Odd = Even. So, this P/2 part can be even or odd, meaning the total result can be odd or even. Not must be even. (Example: 4/2+3 = 2+3=5 (odd); 6/2+3 = 3+3=6 (even)). FALSE
d) (P - Q)^2: (Even - Odd)^2 = (Odd)^2 = Odd * Odd = Odd. (Example: (4-3)^2 = 1^2 = 1). FALSE The correct answer is a) P + Q + 1.
Sum of three consecutive integers:
Let the integers be n, n+1, n+2.
Sum = n + (n+1) + (n+2) = 3n + 3 = 3(n+1).
Since 3(n+1) is always a multiple of 3, the sum is always a multiple of 3.
Regarding parity:
If n is Even: Even + Odd + Even = Odd. (e.g., 2+3+4=9)
If n is Odd: Odd + Even + Odd = Even. (e.g., 1+2+3=6) So, it can be either even or odd. The correct answer is c) A multiple of 3.
If x + y is odd, which must be true?
From the parity rules for addition, Sum is Odd only when Parities are Different.
a) x is even and y is even. (Sum would be Even).
b) x is odd and y is odd. (Sum would be Even).
c) x and y have different parities. TRUE (e.g., one even, one odd).
d) x * y is odd. (This would mean both x and y are odd, which contradicts x+y being odd). The correct answer is c) x and y have different parities.
Parity of (N + 1) N (N - 1):
This expression represents the product of three consecutive integers.
In any set of three consecutive integers, there will always be:
Exactly one multiple of 3.
At least one even number.
Either one even and two odd numbers (if N is odd) OR two even and one odd number (if N is even). Since there is always at least one even number among any three consecutive integers, their product will always contain an even factor. Therefore, the product (N + 1) N (N - 1) is always even. (Furthermore, among any three consecutive integers, at least one is divisible by 2, and exactly one is divisible by 3. This means their product is always divisible by 6. Also, among any three consecutive integers, there's always at least one multiple of 2, and either one multiple of 4 OR another multiple of 2 (if N or N-1/N+1 is a multiple of 4).
Thus, the product is always divisible by 8 if N is even, or by 2 if N is odd and the others are odd. It is always divisible by 2. It is always a multiple of 2, which means it is always even.) The most direct answer is b) It is always even. (It is also always a multiple of 3, and often a multiple of 4 or 8, but "always even" is the most general correct parity statement).
