=
Co-prime numbers
two numbers having no common factor
other than 1 are co-prime numbers.
If a and b are co-prime numbers then they have no common factor other than 1
Example:
12 & 17,
21 & 22,
33 & 40,
Let p be a prime number,
If p divides a2 , then p divides a
Example:
If 2 divides (8)2
then 2 divides 8
If 7 divides (35)2
then 7 divides 35
Proof
Q.1
Let us assume that
is a rational number.
There exist co-prime integers a and b, (b ≠ 0) such that,
∴
=
a
b
Rational number =
a
b
,
(b ≠ 0)
& a, b are co-prime integers
We will prove it by
Contradiction method
5
5
=
a
squaring both sides,
5b2
=
a2
5 divides a2
∴
⇒
5
divides a
... (1)
Let a
=
5c
where c is some integer
... (2)
substituting this value of a in (1)
(5c)2
=
5b2
=
25c2
b2
=
5c2
∴
5b2
5 divides b2
⇒
5 divides b
... (4)
From (3) and (5), we get,
a and b both have common factor 5.
Our assumption that is a rational number is wrong.
∴
This contradicts the fact that a and b are co-prime.
If 5 divides 15
That means 5 × integer = 15
Dividing both side by 5
5 divides a & b both
b
5
5
is an irrational number.
5
Exercise 1.3
5 is a factor of a… 3
After equation 2
5 is a factor of b….5
After equation 4