Скачать или смотреть Question No 01 | Part C | Exercise 4.2 | The Theory of Congruences | Elementary Number Theory

Question No 01 | Part C | Exercise 4.2 | Chapter No 4 | The Theory of Congruences | Elementary Number Theory

Question No 01 | Part B | Exercise 4.2 | The Theory of Congruences Elementary Number Theory

Basic Properties of Congruence | Chapter No 4 | The Theory of Congruences | Elementary Number Theory

Book Name : Elementary Number Theory

By : David M. Burton


Chapter Number : 04
Chapter Name : The Theory of Congruences

Lecture Number : 05
By (Name) : Awais Rasool

Exercise Number : 4.2
Problems Number: 4.2

Question Number : 01
Part Number : C

Example Number : 00
Theorem Number: 4.2
Awais Rasool Shah

Topics Name :
4.1 Carl Friedrich Gauss
4.2 Basic Properties of Congruence.
4.3 Binary and Decimal Representations of integers
4.4 Linear Congruences and the Chinese Remainder Theorem.

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Basic Properties of Congruence:

Definition:
Let 𝑛 be a fixed positive integer. Two integers 𝑎 and 𝑏 are said to be congruent modulo 𝑛, symbolized by
𝑎≡𝑏 (𝑚𝑜𝑑 𝑛)
If 𝑛 divides the difference 𝑎−𝑏; that is, provided that 𝑎−𝑏=𝑘𝑛 for some integer 𝑘.

Let 𝑛 is greathen 1 be fixed and 𝑎,𝑏,𝑐,𝑑 be arbitrary integers. Then the following properties hold:
𝑎≡𝑎 (𝑚𝑜𝑑 𝑛).
If 𝑎≡𝑏 (𝑚𝑜𝑑 𝑛), then 𝑏≡𝑎 (𝑚𝑜𝑑 𝑛).
If 𝑎≡𝑏 (𝑚𝑜𝑑 𝑛) and 𝑏≡𝑐 (𝑚𝑜𝑑 𝑛) then 𝑎≡𝑐 (𝑚𝑜𝑑 𝑛).
If 𝑎≡𝑏 (𝑚𝑜𝑑 𝑛) and 𝑐≡𝑑 (𝑚𝑜𝑑 𝑛) , then 𝑎+𝑐≡𝑏+𝑑 (𝑚𝑜𝑑 𝑛) and 𝑎𝑐≡𝑏𝑑 (𝑚𝑜𝑑 𝑛)
If 𝑎≡𝑏 (𝑚𝑜𝑑 𝑛) , then 𝑎+𝑐≡𝑏+𝑐 (𝑚𝑜𝑑 𝑛) and 𝑎𝑐≡𝑏𝑐 (𝑚𝑜𝑑 𝑛)
If 𝑎≡𝑏 (𝑚𝑜𝑑 𝑛), then 𝑎^𝑘≡𝑏^𝑘 (𝑚𝑜𝑑 𝑛) for any positive integer 𝑘.

Question No: 01
Prove each of the following assertions:
Part: C
If 𝑎≡b (𝑚𝑜𝑑 𝑛) and the integers “𝑎 , 𝑏 , 𝑛” are all divisible
by "d Greater then 0", then 𝑎/𝑑≡𝑏/𝑑 (𝑚𝑜𝑑 𝑛/𝑑).

Sol: Given that
𝑎≡𝑏 (𝑚𝑜𝑑 𝑛)
𝑎−𝑏≡0 (𝑚𝑜𝑑 𝑛)
We can written as:
𝑛|𝑎−𝑏
𝑎−𝑏=𝑘𝑛  (i) if “k” is a constant.
Divided by “d” on both side and “d” is an constant and "d greater then 0".
𝑎/𝑑−𝑏/𝑑=𝑘𝑛/𝑑
𝑎/𝑑−𝑏/𝑑=𝑘/𝑑 𝑛
we can written either 𝑛/𝑑|𝑎/𝑑−𝑏/𝑑 and 𝑘|𝑎/𝑑−𝑏/𝑑 .
we can written either 𝑛/𝑑|𝑎/𝑑−𝑏/𝑑 and 𝑘|𝑎/𝑑−𝑏/𝑑 . .
if 𝑛/𝑑|𝑎/𝑑−𝑏/𝑑 .
𝑎/𝑑−𝑏/𝑑≡0(𝑚𝑜𝑑 𝑛/𝑑)
Add 𝑏/𝑑 both side.
𝑎/𝑑−𝑏/𝑑+𝑏/𝑑≡0+𝑏/𝑑 (𝑚𝑜𝑑 𝑛/𝑑)
𝑎/𝑑≡𝑏/𝑑 (𝑚𝑜𝑑 𝑛/𝑑)
Hence Prove.