The Highest Common Factor (HCF) of 36 and 84 is 12.
The Highest Common Factor (HCF), also commonly known as the Greatest Common Divisor (GCD), is the largest positive integer that divides two or more numbers without leaving a remainder. For 36 and 84, this means finding the greatest integer that can divide both numbers evenly.
Here's how to determine the HCF of 36 and 84 using various methods:
How to Calculate HCF of 36 and 84
There are several effective methods to find the HCF of two numbers like 36 and 84.
1. Listing All Factors
This method involves listing all the factors (divisors) of each number and then identifying the largest factor common to both.
- Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
- Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
Let's visualize this in a table:
| Number | Factors |
|---|---|
| 36 | 1, 2, 3, 4, 6, 9, 12, 18, 36 |
| 84 | 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84 |
By comparing the lists, the common factors are 1, 2, 3, 4, 6, and 12. The highest among these common factors is 12.
2. Prime Factorization Method
The prime factorization method involves breaking down each number into its prime factors. The HCF is then found by multiplying the common prime factors raised to their lowest powers.
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Prime Factorize 36:
- 36 = 2 × 18
- 18 = 2 × 9
- 9 = 3 × 3
- So, the prime factorization of 36 is 2 × 2 × 3 × 3 = 2² × 3²
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Prime Factorize 84:
- 84 = 2 × 42
- 42 = 2 × 21
- 21 = 3 × 7
- So, the prime factorization of 84 is 2 × 2 × 3 × 7 = 2² × 3¹ × 7¹
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Identify Common Prime Factors:
- Both numbers share the prime factors 2 (appearing twice) and 3 (appearing once).
- Common prime factors: 2², 3¹
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Multiply the Common Prime Factors:
- HCF (36, 84) = 2 × 2 × 3 = 12
This method is particularly useful for larger numbers where listing all factors can be cumbersome. For more on prime factorization, you can refer to resources like Khan Academy.
3. Long Division Method (Euclidean Algorithm)
The Euclidean Algorithm is an efficient method for finding the HCF of two numbers by repeatedly dividing the larger number by the smaller number until the remainder is zero. The last non-zero remainder is the HCF.
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Divide 84 by 36:
- 84 = 36 × 2 + 12 (Remainder is 12)
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Now, divide 36 by the remainder (12):
- 36 = 12 × 3 + 0 (Remainder is 0)
Since the remainder is 0, the last non-zero remainder, which is 12, is the HCF of 36 and 84. You can explore more about the Euclidean Algorithm on sites like Math is Fun.
All three methods consistently show that the Highest Common Factor (HCF) of 36 and 84 is 12. This means that 12 is the largest number that divides both 36 and 84 without leaving any remainder.
