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# 数学代写|数论代写Number Theory代考|MATH3320 Definition and Examples of Congruences

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## 数学代写|数论代写Number Theory代考|Definition and Examples of Congruences

Let $n \geq 2$ be a fixed integer. We define two integers $a$ and $b$ to be congruent modulo $n$ if $n$ divides the difference $a-b$. We will denote this by writing $a \equiv b(\bmod n)$. We call the integer $n$ the modulus of the congruence. We note that by the definition of “divides,” $a \equiv b(\bmod n)$ means that $a-b=n k$ for some integer $k$. Note that we require $n$ to be greater than 1 since if $n=1$ then every integer is equivalent to every other integer.

Probably without realizing it, you have already encountered congruences in everyday life. For example, the U.S. clock system works modulo 12 whereas the military clock systems work modulo 24. Days of the week are determined modulo 7 because if a given day is Monday, then seven days later we have another Monday. Similarly, except for leap years, our yearly calendars work modulo 365. Let’s look at some examples.

Example 3.1. (a) We know $27 \equiv 5(\bmod 11)$ since $27-5=22=$ 11(2). Note that 27 is also congruent to 5 modulo 2 since, again, $27-5=22=2(11)$.
(b) One has to be a bit more careful with negative numbers, but the idea is the same. For example, $4 \equiv-21(\bmod 5)$ since $4-(-21)=$ $4+21=25=5(5)$

An alternative way to determine if two integers $a$ and $b$ are congruent modulo $n$ is to use the Division Algorithm to divide each integer by the modulus $n$ and check to see if the two remainders are the same. In Example $3.1$ we noted that $27 \equiv 5(\bmod 11)$. Dividing 27 by 11 we obtain a remainder of 5 , and when dividing 5 by 11 , we also obtain the same remainder of 5 . Because these two remainders are the same, we can conclude that $27 \equiv 5(\bmod 11)$.
Example 3.2. Consider the positive integers 235 and 147 with modulus $n=11$. Dividing 235 by 11 we obtain a remainder of 4; similarly dividing 147 by 11 we also obtain a remainder of 4 . So 235 and 147 are congruent modulo 11. As a check we can also calculate $235-147=88=11(8)$ so that $235 \equiv 147(\bmod 11)$.

## 数学代写|数论代写Number Theory代考|The Finite Sets Zn

The idea of finding the two remainders upon division by the modulus $n$ leads us to an important point which follows directly from the Division Algorithm (Theorem 1.2): Every integer is congruent modulo $n$ to exactly one of $n$ ‘s possible remainders. For each $n>1$, this finite set of remainders, i.e., the set ${0,1, \ldots, n-1}$, turns out to be very important because we can do arithmetic inside this set provided that we do the arithmetic modulo $n$. This set is called the integers $\bmod n$ and is denoted $\mathbb{Z}_n$. To emphasize, we repeat:
$\mathbb{Z}_n={0,1, \ldots, n-1}$ with arithmetic done modulo $n$.
If $a$ is any integer, we shall use the notation $a(\bmod n)$ to denote the unique remainder of $a$ divided by $n$, which is of course an element of $\mathbb{Z}_n$. This remainder is also referred to as the least non-negative residue of $a$ modulo $n$. We shall refer to this operation as reduction $\bmod n$. Note that $” a(\bmod n)$ ” is an object; ” $a \equiv b$ $(\bmod n)$ ” is a statement.

Example 3.3. The least non-negative residue of 27 modulo 5 (which we are denoting as $27(\bmod 5))$ is 2 because, by the Division Algorithm, $27=(5)(5)+2$. Similarly, the least non-negative residue of $-27$ modulo 5 (i.e., $-27(\bmod 5))$ is 3 since $-27=$ $(-6)(5)+3$

## 数学代写数论代写Number Theory代考|Definition and Examples of Congruences

(b) 必须对负数更加小心，但想法是一样的。例如， $4 \equiv-21(\bmod 5)$ 目从 $4-(-21)=4+21=25=5(5)$

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