Understanding HCF: Highest Common Factor

The Highest Common Factor (HCF) of two or more numbers is the largest positive integer that divides each of the numbers exactly without leaving any remainder. It is also sometimes called the Greatest Common Divisor (GCD).

Calculating the HCF of 56, 140, and 168

To find the HCF of 56, 140, and 168, we can use the prime factorization method. This involves breaking down each number into its prime factors.

Step 1: Prime Factorization

Let's find the prime factors for each number:

• For 56:
  • $56 = 2 \times 28$
  • $28 = 2 \times 14$
  • $14 = 2 \times 7$
  So, the prime factorization of 56 is $2 \times 2 \times 2 \times 7 = 2^3 \times 7^1$.
• For 140:
  • $140 = 2 \times 70$
  • $70 = 2 \times 35$
  • $35 = 5 \times 7$
  So, the prime factorization of 140 is $2 \times 2 \times 5 \times 7 = 2^2 \times 5^1 \times 7^1$.
• For 168:
  • $168 = 2 \times 84$
  • $84 = 2 \times 42$
  • $42 = 2 \times 21$
  • $21 = 3 \times 7$
  So, the prime factorization of 168 is $2 \times 2 \times 2 \times 3 \times 7 = 2^3 \times 3^1 \times 7^1$.

Step 2: Identify Common Prime Factors

Now, we look for the prime factors that are common to all three numbers: 56, 140, and 168.

• The prime factors of 56 are $\{2, 7\}$.
• The prime factors of 140 are $\{2, 5, 7\}$.
• The prime factors of 168 are $\{2, 3, 7\}$.

The common prime factors are 2 and 7.

Step 3: Determine the Lowest Power of Each Common Factor

For each common prime factor, we take the lowest power present in the factorizations:

• For the prime factor 2:
  • In 56: $2^3$
  • In 140: $2^2$
  • In 168: $2^3$
  The lowest power of 2 is $2^2$.
• For the prime factor 7:
  • In 56: $7^1$
  • In 140: $7^1$
  • In 168: $7^1$
  The lowest power of 7 is $7^1$.

Step 4: Calculate the HCF

Multiply the lowest powers of the common prime factors together to find the HCF:

$$ \text{HCF}(56, 140, 168) = 2^2 \times 7^1 $$

$$ \text{HCF} = 4 \times 7 $$

$$ \text{HCF} = 28 $$

Thus, the HCF of 56, 140, and 168 is 28.

The common factors with the lowest powers are $2^2$ and $7^1$. Multiplying these gives $4 \times 7 = 28$.

Revision Table: HCF Calculation

Additional Information: HCF and Division Method

Another common method to find the HCF, especially for two numbers, is the division method (Euclidean algorithm). You can extend it to find the HCF of three numbers by first finding the HCF of any two numbers, and then finding the HCF of that result and the third number.

For example, to find HCF(56, 140, 168):

1. Find HCF(56, 140).
   • $140 = 2 \times 56 + 28$
   • $56 = 2 \times 28 + 0$
   HCF(56, 140) is 28.
2. Now find HCF(28, 168).
   • $168 = 6 \times 28 + 0$
   HCF(28, 168) is 28.

So, HCF(56, 140, 168) = 28. This confirms the result obtained using prime factorization.