The Highest Common Factor (HCF) of 84 and 144 is 12. HCF, also known as the Greatest Common Divisor (GCD), is the largest positive integer that divides two or more integers without leaving a remainder. Understanding how to find the HCF is fundamental in number theory and has applications in simplifying fractions and solving mathematical problems.
Methods to Calculate the HCF
There are several effective methods to determine the HCF of two or more numbers. Below, we'll explore three primary approaches, illustrating each with 84 and 144.
1. Listing Factors Method
This method involves listing all the factors (divisors) of each number and then identifying the largest number that appears in both lists.
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Step 1: List all factors for each number.
- Factors of 84: To find the factors of 84, we look for all numbers that divide 84 evenly. These are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84.
- Factors of 144: Similarly, for 144, the factors are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144.
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Step 2: Identify common factors.
- By comparing both lists, the numbers that appear in both are 1, 2, 3, 4, 6, and 12.
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Step 3: Determine the highest common factor.
- From the common factors (1, 2, 3, 4, 6, 12), the highest one is 12.
Therefore, the HCF of 84 and 144 is 12.
Here's a breakdown in a table for clarity:
| Number | Factors |
|---|---|
| 84 | 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84 |
| 144 | 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144 |
| Common Factors | 1, 2, 3, 4, 6, 12 |
2. Prime Factorization Method
This method is efficient, especially for larger numbers, as it breaks down each number into its prime components.
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Step 1: Find the prime factorization of each number.
- For 84: Divide 84 by the smallest prime numbers until only primes remain.
- 84 $\div$ 2 = 42
- 42 $\div$ 2 = 21
- 21 $\div$ 3 = 7
- 7 $\div$ 7 = 1
- So, the prime factorization of 84 is $2 \times 2 \times 3 \times 7 = 2^2 \times 3^1 \times 7^1$.
- For 144:
- 144 $\div$ 2 = 72
- 72 $\div$ 2 = 36
- 36 $\div$ 2 = 18
- 18 $\div$ 2 = 9
- 9 $\div$ 3 = 3
- 3 $\div$ 3 = 1
- So, the prime factorization of 144 is $2 \times 2 \times 2 \times 2 \times 3 \times 3 = 2^4 \times 3^2$.
- For 84: Divide 84 by the smallest prime numbers until only primes remain.
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Step 2: Identify common prime factors and their lowest powers.
- Both numbers share the prime factors 2 and 3.
- For prime factor 2: The lowest power common to both is $2^2$ (from 84).
- For prime factor 3: The lowest power common to both is $3^1$ (from 84).
- Prime factor 7 is only in 84, so it's not common.
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Step 3: Multiply the common prime factors with their lowest powers.
- HCF = $2^2 \times 3^1 = (2 \times 2) \times 3 = 4 \times 3 = 12$.
3. Euclidean Algorithm
The Euclidean Algorithm is a highly efficient method for finding the HCF of two numbers, particularly useful for very large numbers. It's based on the principle that the HCF of two numbers does not change if the larger number is replaced by its difference with the smaller number. This process continues until one of the numbers is zero, and the other number is the HCF. More commonly, it uses division with remainder.
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Step 1: Divide the larger number by the smaller number and find the remainder.
- $144 = 84 \times 1 + 60$ (Here, the remainder is 60)
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Step 2: Replace the larger number with the smaller number, and the smaller number with the remainder.
- Now, we find the HCF of 84 and 60.
- $84 = 60 \times 1 + 24$ (The remainder is 24)
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Step 3: Repeat the process until the remainder is 0.
- Now, we find the HCF of 60 and 24.
- $60 = 24 \times 2 + 12$ (The remainder is 12)
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Step 4: The last non-zero remainder is the HCF.
- Now, we find the HCF of 24 and 12.
- $24 = 12 \times 2 + 0$ (The remainder is 0)
Since the remainder is 0, the last non-zero divisor, which is 12, is the HCF of 84 and 144.
For more information on HCF and GCD, you can explore resources like Wikipedia's page on Greatest Common Divisor or Khan Academy's lessons on HCF.
