HCF of two numbers is 12. Which one of the following can never be their LCM?
80
Understanding HCF and LCM Relationship
The problem asks us to identify which number from the given options can never be the Least Common Multiple (LCM) of two numbers, given that their Highest Common Factor (HCF) is 12.
To solve this, we need to recall a fundamental property that connects the HCF and LCM of any two positive integers. The property states that the LCM of two numbers is always divisible by their HCF. In other words, the LCM is always a multiple of the HCF.
Given that the HCF of the two numbers is 12, their LCM must necessarily be a multiple of 12.
We need to check which of the given options is NOT a multiple of 12.
Checking Each Option's Divisibility by 12
Let's examine each option provided:
| Option | Value | Is it a multiple of 12? | Explanation |
|---|---|---|---|
| 1 | 80 | No | \(80 \div 12\). \(12 \times 6 = 72\) and \(12 \times 7 = 84\). 80 is not exactly divisible by 12. |
| 2 | 60 | Yes | \(60 \div 12 = 5\). 60 is a multiple of 12. |
| 3 | 36 | Yes | \(36 \div 12 = 3\). 36 is a multiple of 12. |
| 4 | 24 | Yes | \(24 \div 12 = 2\). 24 is a multiple of 12. |
Identifying the Impossible LCM
Based on our check, options 2, 3, and 4 (60, 36, and 24) are all multiples of 12. This means they could potentially be the LCM of two numbers whose HCF is 12. For example:
- HCF(12, 60) = 12, LCM(12, 60) = 60.
- HCF(12, 36) = 12, LCM(12, 36) = 36.
- HCF(12, 24) = 12, LCM(12, 24) = 24.
However, option 1 (80) is not a multiple of 12.
Since the LCM of any two numbers must be divisible by their HCF, a number that is not a multiple of the HCF cannot be the LCM.
Therefore, 80 can never be the LCM of two numbers whose HCF is 12.
Conclusion on HCF and LCM Values
The property that LCM is always a multiple of HCF is key here. Any value proposed as an LCM must satisfy this condition. If it doesn't, that value cannot be the LCM for the given HCF.
Revision Table: Key HCF and LCM Facts
| Concept | Definition | Property with LCM |
|---|---|---|
| HCF (Highest Common Factor) | The largest positive integer that divides each of the integers. | LCM is always divisible by HCF. |
| LCM (Least Common Multiple) | The smallest positive integer that is a multiple of both integers. | LCM is always a multiple of HCF. |
Additional Information on HCF and LCM
For any two positive integers 'a' and 'b', the product of their HCF and LCM is equal to the product of the numbers themselves. That is:
$\( \text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b $\)
This relationship also reinforces why the LCM must be a multiple of the HCF. Since \(a \times b\) is divisible by both \(a\) and \(b\), and HCF is a factor of both \(a\) and \(b\), the LCM (which relates to \(a \times b / \text{HCF}\)) will naturally be a multiple of the HCF.
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