The HCF and LCM of two numbers are 9 and 360 respectively. If one number is 45, find the other number.

OR

Show that is irrational, given that is irrational.

Let LCM and HCF of the two numbers be X and Y respectively and let a and b be the two numbers.

According to the given criteria, X = 360 and Y = 9

There is an important formula to remember,

LCM (two numbers) × HCF (two numbers) = Multiplication of two numbers.

Now substituting the corresponding values, we get

X × Y = a × b

⇒ 360 × 9 = a × b

⇒ a × b = 3240

But given one number is 45, so let a = 45, we get

⇒ 45 × b = 3240

⇒ b = 72

Hence the other number is 72.

OR

Given: √5 is irrational

To show: 7 − √5 is irrational

Explanation: we will prove this by contradiction method.

So, let us assume 7 − √5 is rational.

And we know a rational number can be written in form, where a and b are co prime (i.e., a and b have no common factor other than 1) and also b≠0, so

Hence is rational from our assumption.

But given √5 is irrational

So, from equation (i), irrational = rational

But this is not possible, hence this is a contradiction.

So, our assumption is incorrect.

Hence 7 – √5 is irrational.

Hence proved