Prove that √6 is irrational.

1.	Prove that √6 is irrational.
Answer» Let us suppose that √6 is a rational number. There exists two co-prime numbers, say p and q So, √6 = p/q Squaring both sides, we get 6 = p²/q² or 6q² = p² …(1) Which shows that, p² is divisible by 6 This implies, p is divisible by 6 Let p = 6a for some integer a Equation (1) implies: 6q² = 36a² q² = 6a² q² is also divisible by 6 q is divisible by 6 6 is common factors of p and q But this contradicts the fact that p and q have no common factor Our assumption is wrong. Thus, √6 is irrational

Answer»

Let us suppose that √6 is a rational number.

There exists two co-prime numbers, say p and q

So, √6 = p/q

Squaring both sides, we get

6 = p²/q²

or 6q² = p² …(1)

Which shows that, p² is divisible by 6

This implies, p is divisible by 6

Let p = 6a for some integer a

Equation (1) implies: 6q² = 36a²

q² = 6a²

q² is also divisible by 6

q is divisible by 6

6 is common factors of p and q

But this contradicts the fact that p and q have no common factor

Our assumption is wrong. Thus, √6 is irrational

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