To find the Highest Common Factor (HCF) of given numbers, we look for the largest number that divides all the numbers in each pair (or group) without leaving a remainder. We can use either the prime factorization method or the division method.

Let's calculate the HCF for each of the given pairs or groups:

(i) HCF of 4 and 6

• Prime factorization of 4: $2^2$
• Prime factorization of 6: $2 \times 3$
• Common factor: 2
• HCF = 2

(ii) HCF of 9 and 15

• Prime factorization of 9: $3^2$
• Prime factorization of 15: $3 \times 5$
• Common factor: 3
• HCF = 3

(iii) HCF of 30 and 40

• Prime factorization of 30: $2 \times 3 \times 5$
• Prime factorization of 40: $2^3 \times 5$
• Common factors: 2 and 5
• HCF = 10

(iv) HCF of 16 and 20

• Prime factorization of 16: $2^4$
• Prime factorization of 20: $2^2 \times 5$
• Common factor: $2^2$
• HCF = 4

(v) HCF of 6, 12 and 24

• Prime factorization of 6: $2 \times 3$
• Prime factorization of 12: $2^2 \times 3$
• Prime factorization of 24: $2^3 \times 3$
• Common factors: 2 and 3
• HCF = 6

(vi) HCF of 25, 45 and 50

• Prime factorization of 25: $5^2$
• Prime factorization of 45: $3^2 \times 5$
• Prime factorization of 50: $2 \times 5^2$
• Common factor: 5
• HCF = 5

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Would you like further details on any of these? Let me know if you have any questions!

Here are five related questions to expand your understanding:

1. How do you find the Least Common Multiple (LCM) of two numbers?
2. What is the relationship between HCF and LCM?
3. How can we use the Euclidean algorithm to find the HCF?
4. What is the HCF of 18 and 24 using prime factorization?
5. Can the HCF of two numbers ever be larger than the smallest number?

Tip: The HCF is always a divisor of the smallest number in a set, so it can never exceed the smallest number.