Find the HCF and LCM for the following sets of numbers: (i) 24 and 40 (ii) 18, 36 and 75

$$24 = 2 \times 2 \times 2 \times 3 = 2^3 \times 3^1$$ $$40 = 2 \times 2 \times 2 \times 5 = 2^3 \times 5^1$$

The HCF is the product of the common prime factors raised to the lowest power.
Common prime factor is 2. The lowest power of 2 is $2^3$.

$$\text{HCF}(24, 40) = 2^3 = 8$$

The LCM is the product of all prime factors (common and non-common) raised to the highest power.
Prime factors are $2, 3, 5$. Highest power of 2 is $2^3$, 3 is $3^1$, 5 is $5^1$.

$$\text{LCM}(24, 40) = 2^3 \times 3^1 \times 5^1 = 8 \times 3 \times 5 = 120$$

HCF: $\boxed{\text{8}}$
LCM: $\boxed{\text{120}}$

1. (ii) 18, 36 and 75

$$18 = 2 \times 3 \times 3 = 2^1 \times 3^2$$ $$36 = 2 \times 2 \times 3 \times 3 = 2^2 \times 3^2$$ $$75 = 3 \times 5 \times 5 = 3^1 \times 5^2$$

The HCF is the product of the common prime factors raised to the lowest power.
The only common prime factor is 3. The lowest power of 3 is $3^1$.

$$\text{HCF}(18, 36, 75) = 3^1 = 3$$

The LCM is the product of all prime factors (common and non-common) raised to the highest power.
Prime factors are $2, 3, 5$. Highest power of 2 is $2^2$, 3 is $3^2$, 5 is $5^2$.

$$\text{LCM}(18, 36, 75) = 2^2 \times 3^2 \times 5^2 = 4 \times 9 \times 25 = 36 \times 25 = 900$$

HCF: $\boxed{\text{3}}$
LCM: $\boxed{\text{900}}$

1. (iii) 20, 40, 60 and 80

$$20 = 2 \times 2 \times 5 = 2^2 \times 5^1$$ $$40 = 2 \times 2 \times 2 \times 5 = 2^3 \times 5^1$$ $$60 = 2 \times 2 \times 3 \times 5 = 2^2 \times 3^1 \times 5^1$$ $$80 = 2 \times 2 \times 2 \times 2 \times 5 = 2^4 \times 5^1$$

The HCF is the product of the common prime factors raised to the lowest power.
Common prime factors are 2 and 5. The lowest power of 2 is $2^2$. The lowest power of 5 is $5^1$.

$$\text{HCF}(20, 40, 60, 80) = 2^2 \times 5^1 = 4 \times 5 = 20$$

The LCM is the product of all prime factors (common and non-common) raised to the highest power.
Prime factors are $2, 3, 5$. Highest power of 2 is $2^4$, 3 is $3^1$, 5 is $5^1$.

$$\text{LCM}(20, 40, 60, 80) = 2^4 \times 3^1 \times 5^1 = 16 \times 3 \times 5 = 48 \times 5 = 240$$

HCF: $\boxed{\text{20}}$
LCM: $\boxed{\text{240}}$