Prove thatis an irrational number.

OR

Using Euclid’s algorithm, find the HFC of 272 and 1032.

Let √3 be a rational number then it can be written in the form

Where p and q are coprime integers.

Squaring both sides, we get

[1]

⇒ p² = 3q²

⇒ p² has a factor of 3.

⇒ p has a factor of 3 [2]

[If p and q are prime numbers such that q divides p² then q divides p also]

∴ we can write ‘p’ as 3k

From [1], we have

⇒ 3q² = 9k²

⇒ q² = 3k²

⇒ q has a factor of 3

Now, from [2], both p and q have a factor of 3, which is acontradiction to the fact that ‘p’ and ‘q’ are coprime.

∴ our assumption of √3 being a rational number is wrong.

OR

1032 = 272 × 3 + 216

272 = 216 × 1 + 56

216 = 56 × 3 + 48

56 = 48 × 1 + 8

48 = 8 × 6 + 1

∴ HCF(1032, 272) = 8