Question

prove that root 7 is irrationalfind the HCF of 135 and 225 using euclid's division algorithm

Answer 1

Hi

The HCF of 135 and 225 using Euclid division algorithm :

135 and 225

135<225
225 = 135* 1 + 90
the remainder does not become zero, therefore we use again division algorithm . 135>90.

135 = 90*1+45.

the remainder does not become zero. therefore we use again division algorithm.
90>45

90 = 45* 2 + 0

Here the remainder become zero. therefore we stop division algorithm.
the HCF of 135 and 225 is 45.

next we prove that √7 is irrational.

let us assume that √7 is a rational number. then it must be in the form of p/ q where[ q not equal to zero. ]
[p/q are CO prime number.]

√7 = p/q
p= √7*q

squaring both side
p^2 = (√7)^2 * q^2
p2= 7 q2 ----------- (1)equation
p^2 is divisible by 7p and 7q = 7c ( c is a positive integer)

squaring both side

p^2 = 49c^2 ---------------(2)equation

substitute p^2 in 1 equation
7 q^2 = 49 c^2
q^2 = 49/7 c^2
q^2 = 7 c^2
therefore q is divisible by 7 thus p and q have a common factor 7 .
there is a contradiction as our assumptions p and q are CO prime but it has a common factor .
so, it is a irrational number.