For numbers written as products of primes, the HCF is the product of the smallest powers of the common prime factors. The LCM is the product of the greatest powers of all prime factors present.
(i) \(26\) and \(91\)
\[\begin{aligned}26 &= 2\times13,\qquad91\\&= 7\times13.\end{aligned}\]
The only common prime factor is \(13\). Taking the greatest powers of \(2\), \(7\), and \(13\),
\[\begin{aligned}\operatorname{HCF}(26,91) &= 13,\qquad\operatorname{LCM}(26,91)\\&= 2\times7\times13\\&= 182.\end{aligned}\]
Verification:
\[\begin{aligned}\operatorname{HCF}\times\operatorname{LCM} &= 13\times182\\&= 2366,\end{aligned}\]
\[26\times91=2366.\]
Thus, \(\operatorname{HCF}\times\operatorname{LCM}=26\times91\).
(ii) \(510\) and \(92\)
\[\begin{aligned}510 &= 2\times3\times5\times17,\qquad92\\&= 2^2\times23.\end{aligned}\]
The smallest common power of \(2\) is \(2^1\). Taking the greatest powers of all the primes for the LCM,
\[\operatorname{HCF}(510,92)=2,\]
\[\begin{aligned}\operatorname{LCM}(510,92) &= 2^2\times3\times5\times17\times23\\&= 23460.\end{aligned}\]
Verification:
\[\begin{aligned}2\times23460 &= 46920\\&= 510\times92.\end{aligned}\]
Thus, \(\operatorname{HCF}\times\operatorname{LCM}=510\times92\).
(iii) \(336\) and \(54\)
\[\begin{aligned}336 &= 2^4\times3\times7,\qquad54\\&= 2\times3^3.\end{aligned}\]
The smallest common powers are \(2^1\) and \(3^1\). Taking the greatest powers of \(2\), \(3\), and \(7\) for the LCM,
\[\begin{aligned}\operatorname{HCF}(336,54) &= 2\times3\\&= 6,\end{aligned}\]
\[\begin{aligned}\operatorname{LCM}(336,54) &= 2^4\times3^3\times7\\&= 3024.\end{aligned}\]
Verification:
\[\begin{aligned}6\times3024 &= 18144\\&= 336\times54.\end{aligned}\]
Thus, \(\operatorname{HCF}\times\operatorname{LCM}=336\times54\).
Final answer(i) HCF \(=13\), LCM \(=182\)
(ii) HCF \(=2\), LCM \(=23460\)
(iii) HCF \(=6\), LCM \(=3024\)
