FoldUnfold Table of Contents Linear Congruences Linear Congruences In elementary algebra we learn how to solve simple linear equations such as $2x + 3 = 5$. We will now look at solving analogous linear congruences which we define below. Definition: Let $a, b, m \in \mathbb{Z}$. A Linear Congruence is a congruence of the form […]

## Determining the Last Digit of Large Numbers

FoldUnfold Table of Contents Determining the Last Digit of Large Numbers Example 1 Example 2 Determining the Last Digit of Large Numbers Often times it is of great interest to determine the last digit of a very large number. We will now develop a rather simple method involving congruences for doing such. Theorem 1: Let […]

## Tests for Divisibility

FoldUnfold Table of Contents Tests for Divisibility Tests for Divisibility We will now prove some very nice tests for divisibility. Theorem 1: Let $n \in \mathbb{Z}$. Then $n$ is divisible by $9$ if the sum of the digits of $n$ is divisible by $9$. Proof: Let $n \in \mathbb{Z}$ and assume that $n$ is divisible […]

## Example Problems Regarding Congruences

FoldUnfold Table of Contents Example Problems Regarding Congruences Example 1 Example 2 Example 3 Example 4 Example 5 Example 6 Example Problems Regarding Congruences Before we look at some questions regarding congruences, let’s first recall some properties of congruences: 1) $a \equiv b \pmod {m} \implies m \: \mid \: (a – b)$. 2) If […]

## Basic Properties of Congruences Modulo m 2

FoldUnfold Table of Contents Basic Properties of Congruences Modulo m 2 Basic Properties of Congruences Modulo m 2 Recall from the Congruences Modulo m page that if $a, b, m \in \mathbb{Z}$ then $a$ is said to be congruent to $b$ modulo $m$ written $a \equiv b \pmod m$ if: (1) \begin{align} \quad m \mid […]

## Basic Properties of Congruences Modulo m 1

FoldUnfold Table of Contents Basic Properties of Congruences Modulo m 1 Basic Properties of Congruences Modulo m 1 Recall from the Congruences Modulo m page that if $a, b, m \in \mathbb{Z}$ then $a$ is said to be congruent to $b$ modulo $m$ written $a \equiv b \pmod m$ if: (1) \begin{align} \quad m \mid […]

## Congruences Modulo m

FoldUnfold Table of Contents Congruences Modulo m Congruences Modulo m We will now look at a very important and useful type of equivalence relation called the congruence equivalence relation which tells us that two integers are congruent modulo another integer $m$ if they leave the same remainder upon division by $m$. We will begin with […]

## Examples Using Unique Factorization

FoldUnfold Table of Contents Examples Using Unique Factorization Example 1 Example 2 Example 3 Example 4 Example 5 Example 6 Example 7 Examples Using Unique Factorization Before we look at the following examples, let’s first remember some key points: A number p is considered prime if it only has only 1 and p as its […]

## Integers Expressed as a Product of Primes

FoldUnfold Table of Contents Integers Expressed as a Product of Primes Example 1 Example 2 Determining the Greatest Common Factor with Prime Power Decomposition. Example 3 Theorem: An integer n is a mth power if all exponents in its prime power decomposition are divisible by k. Integers Expressed as a Product of Primes Sometimes, it […]

## Theorem 1: Any positive integer can be expressed as a product of primes. The set of primes which expresses this integer as a product is unique.

FoldUnfold Table of Contents Theorem 1: Any positive integer can be expressed as a product of primes. The set of primes which expresses this integer as a product is unique. Theorem 1: Any positive integer can be expressed as a product of primes. The set of primes which expresses this integer as a product is […]

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