The document discusses properties of integers. It defines integers as whole numbers that can be positive, negative or zero. It discusses properties of addition and multiplication of integers like commutativity, associativity, distributivity, identity elements etc. It also discusses absolute value of integers, divisibility, division algorithm, greatest common divisor (GCD) and related theorems. As an example, it calculates the GCD of 2329 and 3151 as 137 and expresses it in the form 3151m + 2329n.

1. TOPIC: PROPERTIES OF INTEGERS
NAME: FARHEEN PATEL
STD : FYBCA
DIV : I
ROLL NO : 29

2. WHAT IS AN INTEGERS?
An integer is a whole number (not a fractional
number) that can be positive, negative, or
zero.
Examples of integers are: -5, 1, 5, 8, 97, and
3,043.
Examples of numbers that are not integers are:
-1.43, 1 3/4, 3.14, .09, and 5,643.1.
The set of integers, denoted Z, is formally
defined as follows:
Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}

3. PROPERTIES OF ADDITION AND
MULTIPLICATION
1.BINARY OPERATION : The operation is called binary
operation if a, b, ∈ Z => a o b ∈ Z.
For e.g. : a, b, ∈ Z => a + b ∈ Z
a, b, ∈ Z => ab ∈ Z
2.COMMUTATIVE PROPERTY : The operation o is
called commutative property if a o b => b o a.
For e.g. : If a, b, ∈ Z then a+b => b+a
If a, b, ∈ Z then ab => ba

4. 3. ASSOCIATIVE PROPERTY : The operation o is called
associative property if (aob)oc = ao(boc).
For e.g. : a, b ∈ Z => (a+b)+c =a+(b+c)
a, b ∈ Z => (a.b).c =a.(b.c)
4. DISTRIBUTIVE PROPERTY : The operation o is called
distributive property if ao(boc)=(aob)o(aoc).
For e.g. : a(b+c)=ab+ac
5. ADDITIVE IDENTITY: Z contains an element o such that ∀a
∈ Z a+o=o+a=a.
6. MULTIPLICATIVE IDENTITY : Z contains an element 1
such that ∀a ∈ Z a.1=1.a=a
7. ADDITIVE INVERSE : For each element a ∈ Z there is an
element b ∈ Z such that a+b=o=b+a.

5. ABSOLUTE VALUE :
 For any integer x , we define absolute value |x| as follows :
|x|= x if x ≥ 0
= x if x ≥ 0
NOTE : 1) For a ∈ Z , |a|≥ 0.
2) For a ∈ Z , |a|=max{a,-a}=|-a|
3) For a ∈ Z , |a|≤ 𝑎 ≤ |a|
Also, -|a|≤ −𝑎 ≤ 𝑎
THEOREMS : 1) |a+b| ≤ |a|+|b| for a, b ∈ Z
Proof:
If a+b≥ 0 , then |a+b|= a+b
≤ |a|+|b|
If a+b < 0 , then |a+b|= -(a+b)
= -a – b
≤ |a|+|b|
∴ 𝑎 + 𝑏 ≤ |𝑎| + |b|.

6. 2)|ab|=|a||b|
If a>0 , b>0 then |ab|= ab
Also, |a||b|= ab
∴ 𝑎𝑏 = 𝑎 𝑏
If a>0 , b<0 then |ab|= -ab
Also, |a||b|= a(-b)= -ab
∴ 𝑎𝑏 = 𝑎 𝑏
If a<0 , b>0 then |ab|= -ab
Also, |a||b|= (-a)b = -ab
∴ 𝑎𝑏 = 𝑎 𝑏

7. DIVISIBILITY IN Z : For integers a, b we say a divides
b if a≠0 and there exists integer c such that b=ac . Notation a|b.
 THEOREMS :
1) If a|b and b|c then a|c where a , b, c ∈ Z
Proof :
Let a|b & b|c
∴ there exists 𝑘1& 𝑘2 such that b= a𝑘1 & 𝑐 = 𝑏𝑘2
∴ 𝑐 = b𝑘2 = 𝑎𝑘1 𝑘2 = 𝑎 𝑘1 𝑘2
∴ 𝑎|𝑐
2) If c|a , c|b then c|ma+nb
Proof:
Since c|a & c|b
∴there exists 𝑘1, 𝑘2 ∈ 𝑍 such that a=c𝑘1 & 𝑏 = 𝑐𝑘2
ma+nb= m(c𝑘1) + 𝑛 𝑐𝑘2 = 𝑐(𝑚𝑘1 + 𝑛𝑘2)
∴ c|ma+nb.

8. DIVISION ALGORITHM:
For given integers a and b with b>0 ,
there exists unique integers q and r such
that
a = bq + r & 0 ≤ 𝑟 < h

9. GREATEST COMMON DIVISOR :
Let a and b be two non- zero integers d is
said to be greatest common divisor of a & b
if 1) d|a & d|b and
2) if c|a & c|b then c|d

10. THEOREM : Any two non-zero integers a and b have a unique
positive g.c.d. This g.c.d can be expressed in the form ma+nb where
m, n∈ Z .
 THEOREM :
If a=bq+r & r ≠ 0 then (a , b) = (b , r)
Proof : Let d = (a , b) & 𝑑1 = 𝑏 , 𝑟
Then d|a & d| b
∴ 𝑑|𝑎 − 𝑏𝑞
∴ 𝑑 | 𝑟
Thus , d divides both b & r
∴ d divides g.c.d of b & r i.e. d | 𝑑1
Now , 𝑑1 = (𝑏 , 𝑟)
∴ 𝑑1 𝑏 & 𝑑1 𝑟
∴ 𝑑1 | 𝑏𝑞 + 𝑟
∴ 𝑑1 | 𝑎
∴ 𝑑1 divides both a & b
∴ 𝑑1 divides g.c.d of a & b i.e. d | 𝑑1 ∵ 𝑑 & 𝑑1 𝑎𝑟𝑒 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑑 = 𝑑1

11. THEOREM : Two non-zero integers a and b are co-
prime if and only if there exists integers m , n ∈ Z such
that ma+nb=1.
 Proof :
If a & b are co-prime then (a , b) = 1
∴ 𝑡ℎ𝑒𝑟𝑒 𝑒𝑥𝑖𝑠𝑡 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠 𝑚 , 𝑛 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑚𝑎 + 𝑛𝑏 = 1
Conversely ,
Suppose that ma+nb = 1 for m , n ∈ Z & let if possible (a , b ) =
d
Thus , d which is common divisor of a & b also divides ma + nb
.
∴ 𝑑 | 𝑚𝑎 + 𝑛𝑏
∴ 𝑑 | 1
As d is positive , we have d = 1 .

12. THEOREM - EUCLID’S LEMMA :
If (a , b ) = 1 & b|ac then b | c .
Proof :
∵ 𝑎 , 𝑏 = 1 𝑡ℎ𝑒𝑟𝑒 𝑒𝑥𝑖𝑠𝑡𝑠 𝑚 , 𝑛 ∈ 𝑍 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡
1 = ma + nb
∵ 𝑐 = 𝑚𝑎𝑐 + 𝑛𝑏𝑐
∵ 𝑏 𝑑𝑖𝑣𝑖𝑑𝑒𝑠 𝑎𝑐 , 𝑏 𝑑𝑖𝑣𝑖𝑑𝑒𝑠 𝑚𝑎𝑐 + 𝑛𝑏𝑐
∵ 𝑏 | 𝑐

13. E.g. : Find the GCD of 2329 & 3151 , Express it in the form
,2329m + 3151n ?
solution :
3151 = 2329 x 1 + 822
2329 = 822 x 2 + 685
822 = 685 x 1 + 137
685 = 137 x 5 + 0
∴ 2329 , 3151 = 137
Now , 137 = 822 – 685 x 1
= 822 – (2329 – 822 x 2 ) x 1
= 822 x 1 – 2329 x 1 + 822 x 2
= 822 x 3 – 2329 x 1
= (3151 – 2329 x 1) x 3 – 2329 x 1
= 3151 x 3 – 2329 x 3 – 2329 x 1
= 3151 x 3 – 2329 x 4
= 3151 x 3 + 2329 (- 4)
∴ 3151𝑚 + 2329𝑛 = 3151 x 3 + 2329 (-4 )