Let the prime factorisation of a be as follows :
a = p1p2 . . . pn, where p1,p2, . . ., pn are primes, not necessarily distinct.
Therefore, a2 = ( p1p2 . . . pn)( p1p2 . . . pn) = p21p22 . . . pn2.
Now, we are given that p divides a2. Therefore, from the Fundamental Theorem of Arithmetic, it follows that p is one of the prime factors of a2. However, using the uniqueness part of the Fundamental Theorem of Arithmetic, we realise that the only prime factors of a2 are p1, p2, . . ., pn. So p is one of p1, p2, . . ., pn.
Now, since a = p1 p2 . . . pn, p divides a.
We are now ready to give a proof that √2 is irrational.
The proof is based on a technique called ‘proof by contradiction’. (This technique is discussed in some detail in Appendix 1).