Suppose, \(\sqrt { 5 } \) is a rational number. Then, \(\sqrt { 5 } \) can be expressed in the form \(\frac{a}{b}\), where a and b are coprime integers and \(b\neq 0\).
\(\therefore \sqrt { 5 } =\frac { a }{ b } \)
On squaring both sides, we get
\(5=\frac { { a }^{ 2 } }{ { b }^{ 2 } } \quad \Rightarrow { a }^{ 2 }=5{ b }^{ 2 }\)     ....(i)
\(\Rightarrow \)  5 divides a².
\(\Rightarrow \)  5 divides a. [by theorem 1]   ...(ii)
So, we can take a = 5m
\(\Rightarrow \)   a²  = 25m² [squaring both sides]
On putting the value of a² in Eq. (i), we get
\(\Rightarrow \)  5 divides b²
\(\Rightarrow \)  5 divides b. [by theorem 1]  ... (iii)
Thus, from Eq.(ii), 5 divides a and from Eq. (iii), 5 divides b. It means 5 is a common factor of a and b. This contradicts that there is no common factor of a and b.
This contradiction arises by assuming that \(\sqrt { 5 } \) is rational.
Hence, \(\sqrt { 5 } \) is irrational number.

We have, (7 x 11 x 13) + 13 = 1001 + 13 = 1014
\(\Rightarrow \) 1014 = 2 x 3 x 13 x 13
We know that, a number is called composite number, if it has atleast one factor other than 1 and the number itself. Here, 1014 is the product of more than two prime numbers, i.e. 2, 3 and 13.
So, it is a composite number.
Now, (7 x 6 x 5 x 4 x 3 x 2 x 1) + 5 = 5045
\(\Rightarrow \) 5045 = 5 x 1009
Thus, it is the product of prime factors 5 and 1009.
Hence, it is also a composite number.

We have, 26 and 91

\(\therefore \)  Prime factors of 26 = 2 x 13
and prime factors of 91 = 7 x 13
Now, LCM of 26 and 91 = 2 x 7 x 13 = 182
and HCF of 26 and 91 = 13
For verification
LCM x HCF = 182 x 13 = 2366
and product of two numbers = 26 x 91 = 2366
Thus, LCM x HCF = Product of two numbers.

Let us assume, to the contrary, that 5 - \( \sqrt{3}\) is rational.
That is, we can find coprime a and b (b \(\neq\)0) such that 5 - \( \sqrt{3}\) = \(\frac{a}{b}\)
Therefore, 5 - \(\frac{a}{b}=\sqrt{3}\)
Rearranging this equation, we get \(\sqrt{3}\) = 5 - \(\frac{a}{b}\)=\(\frac{5b-a}{b}\)
Since a and b are integers, we get 5 - \(\frac{a}{b}\) is rational, and so \( \sqrt{3}\) is rational.
But this contradicts the fact that \(\sqrt{3}\) is irrational.
This contradiction has arisen because of our incorrect assumption that 5 - \( \sqrt{3}\) is rational.
So, we conclude that 5 - \( \sqrt{3}\) is irrational.

Given, HCF (253, 440) = 11 and LCM (253, 440)
= 253 x R
\(\therefore LCM(253,440)=\frac { 253\times 440 }{ HCF(253,440) } \)
\(\Rightarrow 253\times R=\frac { 253\times 440 }{ 11 } \)
R = 40