prove that root p is irrational?

let us assume root p is a rational number then

root p =a/b

squaring on both the sides

p=A square/B square

B square=A square/p

since A square is divisible by p , A is also divisible by p

let us take A=pm then

B square=p2m2/p                                     NOTE; (2 is nothing but square)

B2=pm2

M2=B2/p

since B2 is divisible by p ,B is also divisible by p

since both A and B have common factor as p

my assumsion is wrong

therefore root p is an irrational number

maths oswaal pg no 12 real numbers

question number 5