Find the H.C.F. for the following pairs of numbers: a) 6 and 9, b) 10 and 15, c) 12 and 30, d) 18 and 24, e) 24 and 32

• Factors of 6: $\{1, 2, 3, 6\}$
• Factors of 9: $\{1, 3, 9\}$

• Common factors: $\{1, 3\}$

The H.C.F. is the greatest common factor.
H.C.F. of 6 and 9 is $\boxed{\text{3}}$.

b) 10 and 15

• Factors of 10: $\{1, 2, 5, 10\}$
• Factors of 15: $\{1, 3, 5, 15\}$

• Common factors: $\{1, 5\}$

H.C.F. of 10 and 15 is $\boxed{\text{5}}$.

c) 12 and 30

• Factors of 12: $\{1, 2, 3, 4, 6, 12\}$
• Factors of 30: $\{1, 2, 3, 5, 6, 10, 15, 30\}$

• Common factors: $\{1, 2, 3, 6\}$

H.C.F. of 12 and 30 is $\boxed{\text{6}}$.

d) 18 and 24

• Factors of 18: $\{1, 2, 3, 6, 9, 18\}$
• Factors of 24: $\{1, 2, 3, 4, 6, 8, 12, 24\}$

• Common factors: $\{1, 2, 3, 6\}$

H.C.F. of 18 and 24 is $\boxed{\text{6}}$.

e) 24 and 32

• Factors of 24: $\{1, 2, 3, 4, 6, 8, 12, 24\}$
• Factors of 32: $\{1, 2, 4, 8, 16, 32\}$

• Common factors: $\{1, 2, 4, 8\}$

H.C.F. of 24 and 32 is $\boxed{\text{8}}$.

f) 21 and 28

• Factors of 21: $\{1, 3, 7, 21\}$
• Factors of 28: $\{1, 2, 4, 7, 14, 28\}$

• Common factors: $\{1, 7\}$

H.C.F. of 21 and 28 is $\boxed{\text{7}}$.

4. Let's find the H.C.F. of the following numbers by factorisation method.

a) 9, 12

The common prime factor is $3^1$.
H.C.F. of 9 and 12 is $\boxed{\text{3}}$.

b) 15, 18

The common prime factor is $3^1$.
H.C.F. of 15 and 18 is $\boxed{\text{3}}$.

c) 12, 16

The common prime factor is $2^2$.
H.C.F. of 12 and 16 is $2^2 = \boxed{\text{4}}$.

d) 18, 24

The common prime factors are $2^1$ and $3^1$.
H.C.F. of 18 and 24 is $2 \times 3 = \boxed{\text{6}}$.

e) 36, 48