## 06 Apr If p is a prime then √p is irrational

To prove *√p* is irrational, where p is a prime, we will need the following theorem:

**Theorem:** If is prime then then or

*Proof:* If we are done. So let us assume that Therefore, Hence, by Euclid’s lemma, [Euclid’s lemma: If with then ]

Now to **prove ****√p**** is irrational (where p is a prime):**

Assume that *√p* is rational.

Then there exists two integers a, b such that where gcd(a,b)=1 and

Then Which implies

__________(1)

divides

divides (By the above theorem.)

So there exists an integer such that

So, from (1), we get

(Dividing both sides by p.)

This implies p divides b. This is a contradiction since gcd(a,b)=1.

Thus, when p is a prime, *√**p* is irrational.

**Particular Case:-** *√2* is irrational.

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#### Pankaj Jyoti Mahanta

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