Understanding HCF

The Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), of two or more numbers is the largest positive integer that divides each of the numbers without leaving a remainder. Finding the HCF helps in simplifying fractions and solving various problems in mathematics.

Finding the HCF of 54 and 48

We need to find the Highest Common Factor of the numbers 54 and 48. One common method to find the HCF is by using prime factorization. Let's find the prime factors of each number.

Prime Factorization of 54

To find the prime factors of 54, we divide it by the smallest prime numbers until we are left with a prime number.

• $54 \div 2 = 27$
• $27 \div 3 = 9$
• $9 \div 3 = 3$
• $3 \div 3 = 1$

So, the prime factorization of 54 is $2 \times 3 \times 3 \times 3$, which can be written as $2^1 \times 3^3$.

Prime Factorization of 48

Similarly, we find the prime factors of 48.

• $48 \div 2 = 24$
• $24 \div 2 = 12$
• $12 \div 2 = 6$
• $6 \div 2 = 3$
• $3 \div 3 = 1$

So, the prime factorization of 48 is $2 \times 2 \times 2 \times 2 \times 3$, which can be written as $2^4 \times 3^1$.

Identifying Common Factors

Now, we identify the prime factors that are common to both 54 and 48 and take the lowest power for each common factor.

• The prime factor 2 is present in both factorizations. The lowest power of 2 is $2^1$.
• The prime factor 3 is present in both factorizations. The lowest power of 3 is $3^1$.

The common prime factors with their lowest powers are $2^1$ and $3^1$.

Calculating the HCF

To find the HCF, we multiply these common prime factors with their lowest powers.

HCF $(54, 48) = 2^1 \times 3^1 = 2 \times 3 = 6$.

So, the HCF of 54 and 48 is 6.

Common factors with lowest power: $2^1$ and $3^1$.

HCF = Product of common factors with lowest power = $2^1 \times 3^1 = 6$.

Let's look at the options provided:

• 4
• 7
• 6
• 8

Our calculated HCF is 6, which matches one of the options.

Revision Table: Key Concepts

Additional Information on HCF and LCM

While finding HCF, another related concept is Least Common Multiple (LCM). The LCM is the smallest positive integer that is a multiple of two or more numbers.

Relationship between HCF and LCM

For any two positive integers, say 'a' and 'b', the product of their HCF and LCM is equal to the product of the numbers themselves. That is:

HCF$(a, b) \times$ LCM$(a, b) = a \times b$

Using the numbers 54 and 48 from our problem:

• We found HCF(54, 48) = 6.
• Let's find LCM(54, 48). Using prime factorizations ($54 = 2^1 \times 3^3$, $48 = 2^4 \times 3^1$), the LCM is found by taking the highest power of all prime factors involved.
• LCM$(54, 48) = 2^4 \times 3^3 = 16 \times 27 = 432$.
• Now let's check the relationship: HCF $\times$ LCM = $6 \times 432 = 2592$.
• Product of numbers = $54 \times 48$.

As you can see, HCF $\times$ LCM = $6 \times 432 = 2592$, which is equal to $54 \times 48 = 2592$. This confirms the relationship.