(A) Find the HCF and LCM of 26, 65 and 117, using prime factorisation.

\[\left.\begin{array}{l} 26=13 \times 2 \\ 65=13 \times 5 \\ 117=13 \times 3 \times 3 \end{array}\right\} \]

\[\therefore \mathrm{HCF}=13 \]

\[\mathrm{LCM}=13 \times 2 \times 3 \times 5 \times 3=1170 \]

Let \(\displaystyle \sqrt{2}\) be a rational number.

\[\therefore \sqrt{2}=\frac{\mathbf{p}}{\mathbf{q}} \text {, where } \mathrm{q} \neq 0 \text { and let } \mathrm{p} \text { \& } \mathrm{q} \text { be co-primes. } \]

\[2 q^{2}=p^{2} \Rightarrow p^{2} \text { is divisible by } 2 \Rightarrow p \text { is divisible by } 2 \]

\[\Rightarrow p=2 a \text {, where 'a' is some integer } \]

\[4 a^{2}=2 q^{2} \Rightarrow q^{2}=2 a^{2} \Rightarrow q^{2} \text { is divisible by } 2 \Rightarrow q \text { is divisible by } 2 \]

\[\Rightarrow q=2 b \text {, where 'b' is some integer } \]