How do you find the HCF of a 3 number LCM?

How to Find the Highest Common Factor (HCF) of Three Numbers and Its Relationship with Their Least Common Multiple (LCM)

The question "How do you find the HCF of a 3 number LCM?" is based on a misunderstanding of mathematical terms. The Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), is a property found between two or more numbers. The Least Common Multiple (LCM) of three numbers, however, is a single value. Therefore, you cannot find the HCF of an LCM, as an LCM is just one number.

Instead, this guide will explain how to find the HCF of three numbers and clarify the unique relationship between the HCF and LCM when dealing with a set of three numbers.

What is the Highest Common Factor (HCF)?

The Highest Common Factor (HCF) of a set of numbers is the largest positive integer that divides each of the numbers without leaving a remainder. It is also referred to as the Greatest Common Divisor (GCD). Understanding HCF is fundamental in simplifying fractions, solving problems involving division into equal parts, and various other mathematical applications.

For more details on HCF, you can refer to math resources.

Methods to Find the HCF of Three Numbers

Finding the HCF of three numbers extends the methods used for two numbers. The most common approaches are the Prime Factorization Method and the Long Division Method (Euclidean Algorithm).

1. Prime Factorization Method

This method involves breaking down each number into its prime factors. The HCF is then the product of the common prime factors, each raised to the lowest power it appears in any of the factorizations.

Steps:

1. Find the prime factorization for each of the three numbers.
2. Identify all common prime factors among all three numbers.
3. For each common prime factor, select the lowest power to which it is raised in any of the factorizations.
4. Multiply these lowest powers of the common prime factors to get the HCF.

Example: Find the HCF of 12, 18, and 30.

• Prime Factorization:

  • 12 = $2 \times 2 \times 3 = 2^2 \times 3^1$
  • 18 = $2 \times 3 \times 3 = 2^1 \times 3^2$
  • 30 = $2 \times 3 \times 5 = 2^1 \times 3^1 \times 5^1$

• Common Prime Factors: The common prime factors are 2 and 3.

• Lowest Powers:

  • For prime factor 2: The lowest power is $2^1$ (from 18 and 30).
  • For prime factor 3: The lowest power is $3^1$ (from 12 and 30).

• Calculate HCF: HCF(12, 18, 30) = $2^1 \times 3^1 = 2 \times 3 = 6$.

2. Long Division Method (Euclidean Algorithm)

The Euclidean Algorithm is typically used for two numbers. To find the HCF of three numbers, you can apply it sequentially.

Steps:

1. Find the HCF of any two of the three numbers using the long division method.
2. Find the HCF of the result from step 1 and the third remaining number. This final result is the HCF of all three numbers.

Example: Find the HCF of 12, 18, and 30.

1. Find HCF(12, 18):

   • Divide 18 by 12: $18 = 12 \times 1 + 6$
   • Divide 12 by the remainder 6: $12 = 6 \times 2 + 0$
   • The HCF(12, 18) is 6 (the last non-zero remainder).

2. Find HCF(result from step 1, 30) = HCF(6, 30):

   • Divide 30 by 6: $30 = 6 \times 5 + 0$
   • The HCF(6, 30) is 6.

Therefore, HCF(12, 18, 30) = 6.

The Relationship Between HCF and LCM for Three Numbers

For two numbers, say 'a' and 'b', there is a simple and well-known relationship: HCF(a,b) × LCM(a,b) = a × b. However, this straightforward product relationship does not directly extend to three or more numbers.

For three numbers (let's call them p, q, and r), the relationship between their HCF and LCM is more complex and involves their pairwise HCFs. The formula connecting them is:

$LCM(p,q,r) = \frac{p \times q \times r \times HCF(p,q,r)}{HCF(p,q) \times HCF(q,r) \times HCF(r,p)}$

This formula illustrates a mathematical relationship rather than a direct method to find the HCF solely from the LCM. To use this formula to find any one variable (like HCF(p,q,r)), you would need to know all the other values, including the individual numbers (p, q, r), their LCM, and all their pairwise HCFs (HCF(p,q), HCF(q,r), HCF(r,p)).

This relationship highlights the interconnectedness of these fundamental number theory concepts, even if it doesn't provide a shortcut for finding HCF from a known LCM alone.

Practical Insights and Key Differences

Understanding both HCF and LCM is crucial in various mathematical problems. Here's a quick comparison:

For more on LCM, you can check out online math resources.