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ETS divides the GRE Quantitative reasoning exam into four sections, one of which is Arithmetic. The following is a checklist of the concepts you need to cover to completely prepare for this section. Please note that questions on the GRE do not just check your understanding of the concepts but also your ability to infer or reason using them, an understanding of the concepts is necessary but not the only requirement for preparation.

## Integer Properties & Arithmetic- GRE

Article Index

### Real Numbers, Integers and their Various Classifications

The student:

• knows what real numbers are.
• knows what integers are.
• understands the difference between positive and negative integers.
• understands that 0 is neither a positive nor a negative integer.
• understands the important distinction between the set of positive integers and the set of non-negative integers.
• understands the distinction between the set of negative integers and the set of non-positive integers.
• understands the difference between even and odd integers.
• understands that 0 is an even integer.
• understands the concept of consecutiveness in integers, even integers and odd integers.
• understands the meaning of the term perfect square.

### Divisibility of Integers and Associated Concepts

The student:

• understands what the term divisibility means.
• understands associated terms and phrases such as factor, divisor, “evenly divides”, multiple.
• knows the difference between a factor and a multiple.
• can manually list a few factors and multiples for single and two-digit integers.
• can check the divisibility of an integer by 2, 3, 4, 5, 6, 8 and 9 using shortcuts known as the rules of divisibility.
• knows what a factor pair is.
• can break an integer into a factor pair.
• can find all positive factors of an integer by making a factor pair table (the process of factorization).
• knows what prime numbers are.
• understands that a prime number has only two positive factors: 1 and itself.
• understands that 1 is not a prime number.
• understands that 2 is the only even prime number.
• can break an integer down to its prime factors (prime factorization).
• understands that prime factors are the “building blocks” of an integer.
• understands that all of the positive factors of an integer are really different combinations of its prime factors.
• can calculate the total number of positive factors of an integer through prime factorization.
• can calculate the total number of odd positive factors of an integer.
• can calculate the total number of even positive factors of an integer.
• knows that the product of n consecutive integers is divisible by n.
• can find common factors of two or more integers by manually listing down their positive factors.
• can identify the highest common factor (HCF) of two or more integers by manually listing down their positive factors.
• can find the HCF of two or more integers through prime factorization.
• can find the first few common multiples of two or more integers by manually listing down their positive multiples.
• can identify the lowest common multiple (LCM) of two or more integers by manually listing down their positive multiples.
• can find the LCM of two or more integers through prime factorization.

### Remainder

The student:

• understands the terms divisor, dividend, quotient and remainder.
• knows that the range of possible values of a remainder is between 0 and one less than the divisor, inclusive.
• understands that the remainder is zero only when the divisor is a factor of the dividend (i.e. the divisor evenly divides the dividend).
• knows that the smallest possible positive dividend that, when divided by a divisor, produces a certain remainder is equal to the remainder itself.
• understands that when a multiple of the divisor is added to the dividend, the remainder of the resulting division is the same as that of the original division.
• understands that the dividend can be expressed as the sum of the product of quotient and divisor and the remainder.

### The Arithmetic of Even & Odd Integers

The student:

• knows that the sum of two even integers is an even integer.
• knows that the sum of two odd integers is an even integer.
• knows that the sum of an even integer and an odd integer is an odd integer.
• knows that the multiplication of an even integer to any integer always yields an even integer.
• knows that the product of two odd integers is an odd integer.
• can test whether the result of an algebraic expression is even or odd by using the test values 0 and 1 to represent even and odd input variables respectively.

### Digits

The student:

• knows the distinction between a number and a digit.
• knows and understands the different digit places and their place values on both sides of the decimal point.
• knows how to represent a number as the sum of the place values of its digits.
• understands that the unit digit of a product is determined solely by the unit digits of the multiplying numbers.
• can form patterns of unit digits for successive powers of the same integer and use this pattern to find the unit digits of large powers of integers.

### Representation of Real Numbers on a Number Line, Arithmetic of Real Numbers and Patterns of Successive Integer Powers of Real Numbers

The student:

• knows how to read and interpret a number line.
• understands that the numbers represented by a number line increase when we move from left to right.
• understands that a positive result is obtained when a smaller number is subtracted from a larger number.
• understands that a negative result is obtained when a larger number is subtracted from a smaller number.
• understands that the product or quotient of two negative numbers is a positive number.
• understands that the product or quotient of a positive and a negative number is a negative number.
• understands that the absolute value of the difference of two numbers can be thought of as the distance between those numbers on the number line.
• understands the difference between a proper fraction and an improper fraction.
• can perform basic arithmetic (addition, subtraction, multiplication and division) with fractions
• can calculate integer powers of fractions.
• understands that in terms of patterns of successive integer powers, numbers on a number line fall into four basic categories:
• numbers greater than 1 increase with higher integer powers.
• numbers between 0 and 1 decrease with higher integer powers.
• numbers between -1 and 0 increase with higher odd integer powers, while even integer powers render the result positive.
• numbers less than -1 decrease with higher odd integer powers, while even integer powers render the result positive.