Cheatsheet Odd, Even, Positive, Negative Numbers

1. Odd Numbers

Definition: An odd number is an integer that cannot be divided exactly by 2.

General Form:

2n + 1 (where n is any integer)

Examples:
1, 3, 5, 7, 9, 11 ...

Key Properties:

Odd + Odd = Even
Odd - Odd = Even
Odd + Even = Odd
Odd - Even = Odd
Odd × Odd = Odd
Odd × Even = Even

GMAT Tip: If a number ends in 1, 3, 5, 7, or 9, it is odd.

2. Even Numbers

Definition: An even number is an integer that is divisible by 2.

General Form:

2n (where n is any integer)

Examples:
2, 4, 6, 8, 10, 12 ...

Key Properties:

Even + Even = Even
Even - Even = Even
Even × Even = Even
Even × Odd = Even
Even ÷ Even = Could be even or odd depending on value
Even ÷ Odd = Not an integer unless divisible

GMAT Tip: If a number ends in 0, 2, 4, 6, or 8, it is even.

3. Positive Numbers

Definition: Numbers greater than zero.

Examples:
1, 2, 3, 4, 0.5, 10.8

Key Rules:

(+) × (+) = (+)
(+) ÷ (+) = (+)
(+) × (-) = (-)
(+) ÷ (-) = (-)

Note: Zero is neither positive nor negative, but it is even.

4. Negative Numbers

Definition: Numbers less than zero, always written with a minus sign (-).

Examples:
-1, -2, -3, -0.5, -10.8

Key Rules:

(-) × (-) = (+)
(-) ÷ (-) = (+)
(-) × (+) = (-)
(-) ÷ (+) = (-)

GMAT Tip: Negative numbers are always smaller than positive numbers - even if their absolute values are larger.
For example: -100 is smaller than +1.

5. Important Relationships

Addition/Subtraction

Odd + Odd = Even
Odd + Even = Odd
Even + Even = Even

Multiplication

Odd × Odd = Odd
Odd × Even = Even
Even × Even = Even
Positive × Negative = Negative
Negative × Negative = Positive

Division

Even ÷ Even → Could be even or odd
Even ÷ Odd → May not be integer
Odd ÷ Even → Never an integer

6. GMAT Strategy Notes

1. If a question says "x is an integer," think about odd/even and positive/negative.
2. If two consecutive integers appear in GMAT questions - one must be odd, one must be even.
3. Zero is even but neither positive nor negative.
4. Always check signs carefully in multiplication/division questions.

FAQs on Cheatsheet: Odd, Even, Positive, Negative Numbers

	Quantitative Reasoning for GMAT

1. What are the characteristics of odd numbers?

Ans. Odd numbers are integers that cannot be divided evenly by 2. They are characterised by having a remainder of 1 when divided by 2. Examples include -3, -1, 1, 3, and 5. Odd numbers are important in various mathematical contexts, including number theory and patterns in sequences.

2. How do even numbers differ from odd numbers?

Ans. Even numbers are integers that can be divided evenly by 2, resulting in no remainder. They are represented as 2n, where n is an integer. Examples include -4, -2, 0, 2, and 4. The primary difference between even and odd numbers lies in their divisibility by 2, which influences their behaviours in arithmetic operations and mathematical properties.

3. What defines positive and negative numbers?

Ans. Positive numbers are those greater than zero and are located to the right of zero on the number line. Negative numbers are less than zero and lie to the left of zero. Positive numbers include integers like 1, 2, and 3, while negative numbers include -1, -2, and -3. This distinction is crucial in various mathematical operations, particularly in determining the direction of movement on the number line.

4. How do the properties of positive and negative numbers influence mathematical operations?

Ans. The properties of positive and negative numbers significantly affect arithmetic operations. For instance, adding two positive numbers yields a positive result, while adding two negative numbers results in a negative number. When adding a positive and a negative number, the result's sign depends on the magnitude of the numbers involved. This understanding is essential for solving problems involving integers and real numbers.

5. What are some important relationships involving odd, even, positive, and negative numbers?

Ans. Important relationships include: the sum of two even numbers is even, the sum of two odd numbers is even, and the sum of an odd and an even number is odd. For positive and negative numbers, the sum of two positive numbers is positive, the sum of two negative numbers is negative, and the sum of a positive and a negative number can be either positive or negative, depending on their magnitudes. Understanding these relationships is fundamental for various mathematical problems and examinations.

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Cheatsheet: Odd, Even, Positive, Negative Numbers