# The HCF and LCM o

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) = Product 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

Since, a and b are integers.

7b - a will also be an integer.

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.

Therefore
7 – √5 is irrational.

Hence proved

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