Problem Solving
Quant
7 min readNumber properties are fundamental concepts that help in solving various mathematical problems on the GMAT. This article will cover essential number properties that are crucial for the Problem Solving section, particularly for questions categorized under number properties with a difficulty level of sub-505.
Integers are the set of whole numbers that include all positive numbers, all negative numbers, and zero. They do not include fractions, decimals, or parts of a number. For example, numbers such as –5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5, and so on in both directions are integers. Positive integers (also known as natural numbers) are those greater than zero, while negative integers are those less than zero. Remember, zero is neither positive nor negative.
A rational number is any number that can be written as a fraction where both the numerator and the denominator are integers, with the denominator not equal to zero. This group includes all positive and negative integers, zero, fractions, and decimals that either terminate or have a repeating pattern. For instance, 1/3 is a rational number because its decimal representation is 0.3333… (with the 3 repeating). In contrast, numbers such as √2 are not rational because their decimals neither end nor repeat; these are classified as irrational numbers.
Real numbers encompass all the numbers you typically encounter: integers, rational numbers, and irrational numbers. Every point on a number line represents a real number, whether it is positive, negative, or zero. These numbers are used to measure real-world quantities like weight or temperature. When a problem asks for an answer in terms of real numbers, you can simply use your standard arithmetic or algebraic methods.
Imaginary numbers are numbers that do not fit within the real number system. They are usually written as multiples of the imaginary unit i, where i represents the square root of –1. Because squaring any real number yields a positive result, the square root of a negative number cannot be real—thus, it is called imaginary. Typically, tests such as the GMAT do not require an understanding of imaginary numbers.
Consecutive integers are numbers that follow each other in order, where each number is exactly one more than the previous number. For example, 25, 26, and 27 form a sequence of consecutive integers. They can be all positive, all negative, or a mix, depending on the starting value. A general representation of consecutive integers is n, n+1, n+2, and so forth, where n is any integer.
Prime numbers are defined as positive integers greater than 1 that have exactly two distinct positive factors: 1 and the number itself. Note that 1 is not considered a prime number. The smallest prime is 2, which is unique as the only even prime. It is important to understand that not all odd numbers are prime, and zero is not prime because it is not a positive number and can be divided by any natural number. Examples of prime numbers include 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and so on. Every integer greater than 1 is either prime or can be factored uniquely into a product of prime numbers, such as 15 = 3 × 5 or 100 = 2 × 2 × 5 × 5.
You’re likely already comfortable with the core operations of addition, subtraction, multiplication, and division. However, there are a few key ideas you may want to refresh.
Addition is straightforward—it's just combining two or more numbers to get a total called the sum. For example, if you add 3, 4, and 5, you get 12. Addition also has two key properties:
Subtraction is the opposite of addition. It involves taking one number away from another, resulting in the difference. For example, subtracting 3 from 7 gives 4. Importantly, subtraction doesn’t follow the associative or commutative properties, meaning the order in which you subtract affects the result.
Multiplication can be thought of as repeated addition. For instance, 3 × 5 is the same as adding 5 three times, resulting in 15. Some key points about multiplication:
Division is the reverse of multiplication. It involves splitting a number (the dividend) into equal parts, with the result being the quotient. For example, 15 ÷ 3 equals 5. Division doesn’t follow the commutative or associative properties, and division by zero is undefined.
You probably already know that even numbers are divisible by 2, while odd numbers are not. Here’s what happens when you perform operations with even and odd numbers:
Understanding the behavior of even and odd numbers is vital:
Key properties include:
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. The first few prime numbers are 2, 3, 5, 7, 11, and so on.
Key properties of prime numbers include:
Understanding divisibility is crucial for solving problems efficiently:
Factors are numbers that divide another number without leaving a remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.
Multiples are the result of multiplying a number by an integer. For example, the first few multiples of 3 are 3, 6, 9, 12, and so on.
The absolute value of a number is its distance from zero on the number line, regardless of direction. It is denoted as . For example:
Mastering these number properties is essential for tackling GMAT problems effectively. Practice applying these concepts to various problems to enhance your problem-solving skills.