(i) The numbers are 26 and 91.
Prime factorization of 26 is \(2 \times 13\).
Prime factorization of 91 is \(7 \times 13\).
The HCF is \(13\).
The LCM is \(2 \times 7 \times 13 = 182\).
The product of the two numbers is \(26 \times 91 = 2366\).
The product of HCF and LCM is \(13 \times 182 = 2366\).
Therefore, the product of two numbers equals the product of their HCF and LCM.
(ii) The numbers are 510 and 92.
Prime factorization of 510 is \(2 \times 3 \times 5 \times 17\).
Prime factorization of 92 is \(2 \times 2 \times 23\).
The HCF is \(2\).
The LCM is \(2 \times 2 \times 3 \times 5 \times 17 \times 23 = 23460\).
The product of the two numbers is \(510 \times 92 = 46920\).
The product of HCF and LCM is \(2 \times 23460 = 46920\).
Therefore, the product of two numbers equals the product of their HCF and LCM.
(iii) The numbers are 336 and 54.
Prime factorization of 336 is \(2 \times 2 \times 2 \times 2 \times 3 \times 7\), which is \(2^4 \times 3 \times 7\).
Prime factorization of 54 is \(2 \times 3 \times 3 \times 3\), which is \(2 \times 3^3\).
The HCF is \(2 \times 3 = 6\).
The LCM is \(2^4 \times 3^3 \times 7 = 3024\).
The product of the two numbers is \(336 \times 54 = 18144\).
The product of HCF and LCM is \(6 \times 3024 = 18144\).
Therefore, the product of two numbers equals the product of their HCF and LCM.