2.Euclid's division lemma-Given positive integers a and b , there exist unique integers q and r satisfying a=bq+r,0 less than or equal to r <b.
3. Fundamental theorem of arithmetic-Every composite number can be expressed (factorised) as aproduct of primes, and this factorisation is unique, apart from the order in which the prime factors occur...
5*6*2*3+3=183 which is a composite number .
rearranging this equation we get√5=1/2-a/b=b-2a/b
since a and b are integers , we get 1/2-a/b is rational and so √5 is rational.
but this contradicts the fact that √5 is irrational . This contradiction has has arisen because of our incorrect assumption that 1/2-√5 is rational . so we conclude that 1/2-√5 is irrational.