Question

Prove that square root 31 is irrational.

Answer 1

let us assume that √31 be rational.
then it must in the form of p / q [q ≠ 0] [p and q are co-prime]

√31 = p / q

=> √31 x q = p

squaring on both sides

=> 31q²= p² ------> (1)

p² is divisible by 31

p is divisible by 31

p = 31c [c is a positive integer] [squaring on both sides ]

p² = 961 c² --------- > (2)

subsitute p² in equ (1) we get

31q² = 961 c²

q² = 31c²

=> q is divisble by 31

thus q and p have a common factor 31.

there is a contradiction

as our assumsion p & q are co prime but it has a common factor.

so that √31 is an irrational.