Choose the correct statement from the following.
HCF of two or more numbers is the highest number which perfectly divides all the given numbers.
Understanding the Highest Common Factor (HCF) Definition
The question asks us to identify the correct statement regarding the Highest Common Factor (HCF) of numbers. Let's analyze each option to understand the definition and properties of HCF.
Analyzing the HCF Statements
Let's examine each statement given in the options:
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Statement 1:
HCF is the least common multiple of the given numbers.
This statement is incorrect. HCF and Least Common Multiple (LCM) are distinct concepts. HCF is about the largest factor common to the numbers, while LCM is about the smallest multiple common to the numbers.
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Statement 2:
HCF of two or more numbers is the highest number which perfectly divides all the given numbers.
This statement provides the fundamental definition of HCF. The HCF (also known as the Greatest Common Divisor or GCD) of two or more integers is the largest positive integer that divides each of the integers without leaving a remainder.
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Statement 3:
HCF is also called the least common divisor.
This statement is incorrect. HCF is the *highest* common divisor, not the least. The least common divisor of any two positive integers is always 1, as 1 divides all integers.
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Statement 4:
In prime factorisation method of HCF, the multiples of all the given numbers are listed.
This statement is incorrect regarding the HCF calculation using prime factorization. The prime factorization method for HCF involves finding the prime factors of each number and then multiplying the lowest powers of the common prime factors. Listing multiples is a method typically associated with finding the Least Common Multiple (LCM), not HCF.
Identifying the Correct HCF Definition
Based on the analysis of each statement, the correct definition of HCF is given in Statement 2.
Let's consider an example. Find the HCF of 12 and 18.
- Factors of 12: 1, 2, 3, 4, 6, 12
- Factors of 18: 1, 2, 3, 6, 9, 18
The common factors are 1, 2, 3, and 6. The highest among these common factors is 6. So, HCF(12, 18) = 6. Notice that 6 is the highest number that perfectly divides both 12 and 18.
Conclusion on HCF Definition
The statement that accurately defines the HCF is that it is the highest number which perfectly divides all the given numbers.
| Concept | Definition | Method Example (for 12 & 18) |
|---|---|---|
| HCF (Highest Common Factor) | Largest number dividing all given numbers perfectly. | Common factors are 1, 2, 3, 6. Highest is 6. HCF(12, 18) = 6. |
| LCM (Least Common Multiple) | Smallest number that is a multiple of all given numbers. | Multiples of 12: 12, 24, 36, ... Multiples of 18: 18, 36, 54, ... Least common multiple is 36. LCM(12, 18) = 36. |
| Prime Factorization (for HCF) | Find prime factors, multiply lowest powers of common factors. | $12 = 2^2 \times 3^1$, $18 = 2^1 \times 3^2$. Common factors are $2^1$ and $3^1$. HCF $= 2^1 \times 3^1 = 6$. |
Revision Table: HCF and Related Terms
| Term | Description | Relation to HCF |
|---|---|---|
| Factor | A number that divides another number perfectly. | HCF is the highest among common factors. |
| Common Factor | A number that is a factor of two or more numbers. | HCF is the largest common factor. |
| Divisor | Another name for a factor. | HCF is the Greatest Common Divisor (GCD). |
| Multiple | The result of multiplying a number by an integer. | Distinct from HCF; related to LCM. |
| Least Common Multiple (LCM) | The smallest positive integer that is a multiple of two or more numbers. | Conceptually different from HCF. |
Additional Information on HCF Calculation
Besides listing factors, the HCF can be found using other methods:
- Prime Factorization Method: Find the prime factorization of each number. The HCF is the product of the lowest powers of all common prime factors.
- Euclidean Algorithm: An efficient method for finding the HCF of two numbers 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, or more efficiently, by the remainder when the larger number is divided by the smaller number.
Understanding the definition of HCF is crucial for solving problems involving factors and divisors of numbers.
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Important Questions from LCM and HCF
The greatest three-digit number which is divisible by 14, 28, and 42 is:
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