3. Euclidโs division Algorithm
๏ด Euclidโs division Algorithm โ
๐๐ ๐๐๐ก๐๐๐ ๐กโ๐ ๐ป๐ถ๐น ๐๐ ๐ก๐ค๐ ๐๐๐ ๐๐ก๐๐ฃ๐ ๐๐๐ก๐๐๐๐๐ ๐ ๐๐ฆ ๐ ๐๐๐ ๐ ๐ค๐๐กโ ๐
> ๐, ๐๐๐๐๐๐ค ๐กโ๐ ๐ ๐ก๐๐๐ ๐๐๐๐๐ค โถ
1. Apply Euclidโs division lemma to ๐ and ๐ . So , we find whole numbers , ๐ ๐๐๐ ๐
such that ๐ = ๐๐ + ๐, 0 โค ๐ < ๐.
2. If ๐ = 0 , d is the HCF of ๐ and ๐ . If ๐ โ 0 apply the division lemma to ๐ and ๐ .
3. Continue the process till the remainder is zero . The divisor at this stage will be the
required HCF .
4. Example :- Using Euclidโs division algorithm find the HCF of 12576 and 4052
.
Since 12576 > 4052 we apply the division lemma to 12576 and 4052 to get
12576 = 4052 ร 3 + 420
Since the remainder 420 โ 0 , we apply the division lemma to 4052 and 420 to get
4052 = 420 ร 9 + 272
We consider the new divisor 420 and new remainder 272 apply the division lemma to get
420 = 272 ร 1 + 148
Now we continue this process till remainder is zero .
272 = 148 ร 1 + 124
148 = 124 ร 1 + 24
124 = 24 ร 5 + 4
24 = 4 ร 6 + 0
The remainder has now become 0 , so our procedure stops . Since the divisor at this stage is 4 ,
the HCF of 12576 and 4052 is 4 .
5. Fundamental Theorem of Arithmetic
๐ธ๐ฃ๐๐๐ฆ ๐๐๐๐๐๐ ๐๐ก๐ ๐๐ข๐๐๐๐ ๐๐๐ ๐๐ ๐๐ฅ๐๐๐๐ ๐ ๐๐ ๐๐ ๐ ๐๐๐๐๐ข๐๐ก ๐๐ ๐๐๐๐๐๐ , ๐๐๐
๐กโ๐๐ ๐๐๐๐ก๐๐๐๐ ๐๐ก๐๐๐ ๐๐ ๐ข๐๐๐๐ข๐ , ๐๐๐๐๐ก ๐๐๐๐ ๐กโ๐ ๐๐๐๐๐ ๐๐ ๐คโ๐๐โ ๐กโ๐๐ฆ ๐๐๐๐ข๐.
Now factorise a large number say 32760
2 32760
2 16380
2 8190
3 4095
3 1365
5 455
7 91
13 13
6. Revisiting Irrational Numbers
๐ฟ๐๐ก ๐ ๐๐ ๐ ๐๐๐๐๐ ๐๐ข๐๐๐๐ ๐๐ ๐ ๐๐๐ฃ๐๐๐๐ ๐2, ๐กโ๐๐ ๐ ๐๐๐ฃ๐๐๐๐ ๐ , ๐ ๐๐ ๐ ๐๐๐ ๐๐ก๐๐ฃ๐ ๐๐๐ก๐๐๐๐
Theorem - 2 ๐๐ ๐๐๐๐๐ก๐๐๐๐๐.
proof
Let us assume on contrary that 2 is rational where a and b are co-prime .
โ 2 =
๐
๐
(๐ โ 0)
squaring on both sides
2
2
=
๐
๐
2
๐2
=
๐2
2
Here 2 divides ๐2
, so it also divides ๐ .
so we can write a=2c for some integer c .
7. Substituting for ๐
we get
2๐2
= 4c2
๐2
= 2c2
๐2
=
๐2
2
Here 2 divides ๐2 , so it also divides ๐ .
This creates a contradiction that a and b have no common factors other than 1 .
This contradiction has arisen because of our wrong assumption .
So we conclude that 2 is a irrational number .