Odd and Even Numbers (GMAT Quantitative Section)

MKS Education • Quantitative Reasoning

1) Definition

Even Number: An integer divisible by 2.

n = 2k, where k is an integer.
Examples: …, −4, −2, 0, 2, 4, 6, 8, …

Odd Number: An integer not divisible by 2, leaving a remainder of 1.

n = 2k + 1, where k is an integer.
Examples: …, −3, −1, 1, 3, 5, 7, 9, …

2) Basic Properties

Tip: Multiplying by an even number always gives an even result, since one factor includes 2.

3) Quick Facts

• Zero (0) is even because 0 = 2 \times 0.
• The product of any group of numbers containing one even number is even.
• The difference between two even or two odd numbers is even.
• Squares preserve parity:
  • (2k)^2 = 4k^2 → Even
  • (2k + 1)^2 = 4k^2 + 4k + 1 → Odd
• Consecutive integers n and n+1 have opposite parity, so n(n+1) is always even.

4) Parity in Algebraic Expressions

• If n is even, n + 1 is odd.
• If n is odd, n + 1 is even.
• (\text{even})^m is even; (\text{odd})^m is odd.
• The product n(n+1) is always even (one factor is even).

5) GMAT-Level Applications

1. If a is even and b is odd, find the parity of a^2 + b.
   a^2 is even; even + odd = odd.
2. Is \frac{n(n+3)}{2} an integer when n is odd?
   Let n = 2k + 1: (2k+1)(2k+4) = 2(2k+1)(k+2) → divisible by 2 → Yes.
3. Sum of first n integers: 1 + 2 + 3 + \dots + n = \frac{n(n+1)}{2}.
   Since n(n+1) is even, the sum is always an integer.

6) Summary Table