(i) For 12, 15, and 21:
Prime factorization:
$12 = 2^2 \times 3$
$15 = 3 \times 5$
$21 = 3 \times 7$
HCF is the product of the common prime factors with the lowest powers:
$HCF(12, 15, 21) = 3$
LCM is the product of all the prime factors, each raised to the highest power:
$LCM(12, 15, 21) $
$= 2^2 \times 3 \times 5 \times 7 = 420$
(ii) For 17, 23, and 29:
Since 17, 23, and 29 are all prime numbers, their prime factorizations are:
$17 = 17$
$23 = 23$
$29 = 29$
HCF is the product of the common prime factors:
$HCF(17, 23, 29) = 1$ (since there are no common prime factors)
LCM is the product of all prime factors:
$LCM(17, 23, 29) $
$= 17 \times 23 \times 29 = 11131$
(iii) For 8, 9, and 25:
Prime factorization:
$8 = 2^3$
$9 = 3^2$
$25 = 5^2$
HCF is the product of the common prime factors with the lowest powers:
$HCF(8, 9, 25) = 1$ (since there are no common prime factors)
LCM is the product of all prime factors, each raised to the highest power:
$LCM(8, 9, 25) $
$= 2^3 \times 3^2 \times 5^2 = 1800$