GRE Math Arithmetic | Quantitative Reasoning for GRE PDF Download

Integers

Integers are numbers that include 0, positive numbers, and negative numbers. The positive integers are greater than 0, negative integers are less than 0, and 0 is neither positive nor negative. Integers don’t include fraction, decimal, or percentage. Integers in the GRE Arithmetic represent the numbers {.....-3,-2,-1,0,1,2,3….}.

Different forms of integers with mathematical operations are as follows:

• Addition of two integers will be an integer (negative or positive)
• Subtraction of two integers will be an integer (negative or positive)
• The product of two positive integers is a positive integer
• The product of two negative integers is a positive integer
• The product of a positive integer and a negative integer is a negative integer
• The division of two integers can be an integer or decimal (negative or positive)
• The square of an integer (negative or positive) will be a positive integer
• The root of a positive integer can be an integer or decimal

Some of the concepts you need to master to solve the integer in GRE arithmetic are:

• Factor or Divisor: When integers are multiplied, each multiplied integer is called a factor or divisor.
• Least Common Multiple: LCM of two nonzero integers a and b is the least positive integer that is a multiple of both a and b.
• Greatest Common Factor/Divisor: GCD/GCF of two nonzero integers a and b is the greatest positive integer that is a divisor of both a and b.
• Even and Odd Integer: If an integer is divisible by 2, it is called an even integer otherwise it is an odd integer.
• Prime Number: A prime number is an integer greater than 1 and can be divisible by two positive integers 1 and the integer itself. The first ten prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.
• Prime Factorization: Prime factorization is the process of expressing a composite number as a product of its prime factors. This means finding the prime numbers that multiply together to give the original number.

Example: If x and y are positive integers such that x × y = 72x, what is the greatest possible value of x+y?
Ans: To find x + y, we need to choose x and y to maximize their sum while their product remains 72. The pairs (x, y) that multiply to 72 are:

• (1, 72): x+y=1+72=73
• (2, 36): x+y=2+36=38
• (3, 24): x+y=3+24=27
• (4, 18): x+y=4+18=22
• (6, 12): x+y=6+12=18
• (8, 9): x+y=8+9=17

The maximum sum x+y occurs with the pair (1, 72), giving x + y = 73.

Fractions

A fraction is a number in the form of a numerator/denominator, where the numerator and denominator are integers and the denominator is ≠ 0. For example: 7/8 is a fraction in which 7 is the numerator and 8 is the denominator. These numbers are also called rational numbers.
A numerical expression such as 2⅘ is called a mixed number that contains an integer part and a fraction part. The fraction in the GRE can be done with arithmetic operations such as addition, subtraction, multiplication, and division.

• If the denominators of two or more fractions are the same, simply add their numerators.
• If the denominators of two or more fractions are different, then find the common denominator, and make them equal by multiplying so that the denominators become the same. And, multiply the numerator by the same number too.
• To multiply two fractions, multiply the two numerators by the two denominators.
• To divide fractions, multiply the dividend by the reciprocal of the divisor.

Exponent and Roots

Exponents are used to represent the repeated multiplication of a number by itself. They indicate the number of times a number (the base) is multiplied by itself. For example, in aⁿ  - a is the base and n is the exponent. To practice the GRE exponents, you need to understand the following rules.

• If the exponent is 2, it is known as squaring i.e. 6² and if the exponent is 3, it is known as cubed i.e. 6³.
• If a negative number is raised to powers, the result will be either positive or negative.
• Exponents can also be zero or negative.

Mathematical Operations on Exponents are as follows:

• Multiplication: a^m × aⁿ = a^{m + n}
• Division: a^m ÷ aⁿ = a^{m ÷ n}
• Power of a power: (a^m)ⁿ = a^{m x n}
• Zero exponent: Any non-zero base raised to the power of zero is 1 (a⁰ = 1)
• Negative exponent: (a)^-x = (1/a)^x

A square root of a number is a value when multiplied by itself, giving the original number. A square root is a root of order 2. Similarly, the cube root and fourth root are denoted by the orders 3 and 4.

• Every positive number has two square roots, one positive and one negative (e.g., square roots of 25 are 5 and -5).
• The non-negative square root is represented by √ . For instance, √25=5.
• The square root of 0 is 0.
• Square roots of negative numbers do not have a real number system. 

Rules regarding operations with roots are as follows:

• Rule 1: (√a)² = a
• Rule 2: √a² = a
• Rule 3: √a √b = √ab
• Rule 4: √a / √b = √a/b

Decimals

A decimal is a number that consists of a whole number and a fraction. The decimal number system represents numbers using powers of 10.
Key points about decimals in GRE arithmetic:

• The first digit to the right of the decimal point is the tenth place.
• The second digit is the hundredth place.
• The third digit is the thousandth place, and so on.
• Every fraction with integers in the numerator and denominator is equivalent to a decimal that terminates or repeats. 
• Every rational number can be expressed as a terminating or repeating decimal.
• Two decimals that are not terminating or repeating are known as irrational numbers.

Real Numbers

Real Numbers consist of all rational numbers and irrational numbers. The real numbers include all integers, fractions, and decimals. The set of real numbers can be represented by a number line which is known as a real number line. In the number line, all numbers to the left of 0 are negative and all numbers to the right of 0 are positive. Only the number 0 is neither negative nor positive. 

Key points to be considered in GRE arithmetic real numbers are:

• A real number ‘a’ is less than a real number ‘b’ if a is to the left of b on the number line i.e. a < b.
• A real number ‘a’ is greater than ‘b’ if ‘a’ is to the right of the ‘b’ on the number line i.e. a > b
• The set of all real numbers that are between 2 and 3 is called an interval, and the double inequality 2 < a < 3 is often used to represent that interval.
• The entire real number line is also considered an interval.
• The distance between a number x and 0 on the number line is called the absolute value of x i.e. |x|.

Ratios

A ratio is a comparison of two or more numbers that indicate relative sizes, in which the first number is the numerator and the second number is the denominator. Ratios can be in the form of fractions. The proportion is an equation relating two ratios. To solve the GRE ratio problems, it is preferable to write a proportion and then do cross multiplication.

Percent

A percent is a number or ratio that is used to represent parts of a whole, where a whole is considered as having 100 parts. The part is the numerator of the ratio and the whole is the denominator. Percent is denoted as %. 

Key things to be known in the arithmetic percent are:

• When a quantity changes from an initial positive amount to another positive amount, it is known as percent change.
• If a quantity increases, then it is known as percent increase.
• If a quantity decreases, then it is known as percent decrease.
• When computing a percent increase, the base is the smaller number. 
• When computing a percent decrease, the base is the larger number.

What is the order of operations in GRE arithmetic?

The order of operations in GRE Arithmetic is represented as PEMDAS (Parentheses Exponents Multiplication Division Addition Subtraction). 
When solving an expression with operations of the same type (e.g., only addition, subtraction, multiplication, or division), you should work from left to right. However, for expressions involving multiple arithmetic operations, you need to follow the order of operations.
The order of operations is a rule that suggests the sequence for solving expressions with different operations. You can remember this order with the acronym PEMDAS:

• Parentheses: First, solve the operations within parentheses or brackets. 
• Exponents: Next, evaluate any exponential expressions.
• Multiplication and Division: Then, moving from left to right, perform all multiplication and division operations, in the order they appear.
• Addition and Subtraction: Finally, moving from left to right, complete all addition and subtraction operations, in the order they appear.

Following the order is important especially when putting information into the calculator.