Lesson 6.1: Properties and Types of Integers

Introduction

In this lesson, we will explore the properties and types of integers, which are fundamental in understanding arithmetic concepts tested on the GRE. The objectives for this lesson include:

Understanding divisibility, factorization, prime numbers, and remainders.
Recognizing odd and even integers and their behavior under various operations.
Utilizing integer properties to simplify problems efficiently.
Applying divisibility and factorization rules to solve arithmetic problems.
Analyzing remainders and prime factorization.

By the end of this lesson, students will have a solid grasp on these concepts, enabling effective problem-solving techniques applicable to the GRE quantitative reasoning section.

Properties of Integers

Definition of Integers

Integers are whole numbers that can be positive, negative, or zero. They can be represented as:

$$ Z = \{ \ldots, -3, -2, -1, 0, 1, 2, 3, \ldots \} $$

Types of Integers

Even Integers: An integer is considered even if it is divisible by 2. Mathematically, an integer $n$ is even if:

$$ n = 2k $$

for some integer $k$.

Example: The integers $-4, -2, 0, 2, 4, 6$ are all even.

Odd Integers: An integer is odd if it is not divisible by 2. An integer $n$ is odd if:

$$ n = 2k + 1 $$

for some integer $k$.

Example: The integers $-3, -1, 1, 3, 5$ are all odd.

Divisibility

Divisibility indicates whether one integer can be divided by another without leaving a remainder. An integer $a$ is divisible by an integer $b$ if there exists an integer $k$ such that:

$$ a = b \cdot k $$

For instance, since $10 = 2 \cdot 5$, we say $10$ is divisible by $2$.
A common way to express this is $b | a$. For example, $2 | 10$ means $2 divides $10$.

Example of Divisibility

Determine if $12$ is divisible by $3$.

Find $k$ such that $12 = 3 \cdot k$.
Here, $k = 4$ (since $12 = 3 \cdot 4$), thus $12$ is divisible by $3$.

Remainders

When one integer is divided by another, the remainder is what is left over after performing the division. For integers $a$ and $b$ (where b

eq 0), we can express this as:

$$ a = b \cdot q + r $$

where $q$ is the quotient and $r$ (the remainder) satisfies $0 \leq r < b$.

Example of Remainder

What is the remainder when $29$ is divided by $5$?

Performing the division, we have:

$$ 29 = 5 \cdot 5 + 4 $$

Therefore, the remainder $r = 4$.

Prime Numbers

Definition

A prime number is an integer greater than $1$ that has no positive divisors other than $1$ and itself. In other words, a prime number $p$ cannot be written as:

$$ p = a \cdot b $$

where both $a$ and $b$ are integers greater than $1$.

Examples of Prime Numbers

The first few prime numbers are $2, 3, 5, 7, 11, 13, 17, 19, 23, 29. Note that $2$ is the only even prime number; all other even integers are divisible by $2$ and thus cannot be prime.

Prime Factorization

Every integer greater than $1$ can be expressed as a product of prime numbers. This representation is unique up to the order of the factors and is known as prime factorization.

Example of Prime Factorization

Find the prime factorization of $60$.

Start with $60$. Since it is even, divide by $2$:

$60 ÷ 2 = 30$

Divide $30$ by $2$:

$30 ÷ 2 = 15$

Since $15$ is not even, divide by the next prime, $3$:

$15 ÷ 3 = 5$

$5$ is a prime number.

Thus, the prime factorization of $60$ is:

$$ 60 = 2^2 \cdot 3^1 \cdot 5^1 $$

Odd and Even Operations

Properties of Odd and Even Integers

Even + Even = Even: e.g., $2 + 4 = 6$
Odd + Odd = Even: e.g., $3 + 5 = 8$
Even + Odd = Odd: e.g., $2 + 3 = 5$
Even × Even = Even: e.g., $2 \cdot 4 = 8$
Odd × Odd = Odd: e.g., $3 \cdot 5 = 15$
Even × Odd = Even: e.g., $2 \cdot 3 = 6$

Example of Operations

If we have $4$ (even) and $5$ (odd), consider the operation:

$4 + 5 = 9$ (odd)
$4 \cdot 5 = 20$ (even)
If we add $4$ to itself, $4 + 4 = 8$ (even)
What about subtracting: $5 - 4 = 1$ (odd)

Conclusion

Understanding the properties and types of integers provides a foundation for tackling more complex arithmetic problems on the GRE. Recognizing the characteristics of odd and even integers, learning the rules of divisibility, and mastering prime factorization will be invaluable. students should practice applying these principles to become proficient in recognizing and solving problems quickly.

Study Notes

Integers include positive, negative numbers, and zero.
Even integers are of the form $2k$, and odd integers are of the form $2k + 1$.
An integer $a$ is divisible by $b$ if there exists an integer $k$ such that $a = b \cdot k$.
The remainder from the division $a \div b$ can be expressed as $a = b \cdot q + r$ where $0 \leq r < b$.
A prime number is only divisible by $1$ and itself.
Every integer greater than $1$ can be expressed uniquely as a product of prime factors.
Operations with even and odd integers yield predictable results.