Let L be the LCM and H be the HCF of two given numbers. L and H are in the ratio 3 ∶ 2. If the sum of the two numbers is 45, then what is the product of the numbers?
Cannot be determined due to insufficient data
Understanding the HCF, LCM, Sum, and Product
This problem asks us to find the product of two numbers given information about their Highest Common Factor (HCF), Least Common Multiple (LCM), and sum. Let the two numbers be \(a\) and \(b\). We are given the ratio of their LCM (\(L\)) and HCF (\(H\)) and their sum.
Key Relationships Between HCF and LCM
There are fundamental relationships between two positive integers, their HCF, and their LCM:
- The product of two numbers is equal to the product of their HCF and LCM. That is, \(a \times b = H \times L\).
- The HCF of two numbers always divides their LCM exactly. This means \(\frac{L}{H}\) must be an integer.
- If the two numbers are expressed as \(a = Hx\) and \(b = Hy\), where \(x\) and \(y\) are integers, then \(x\) and \(y\) must be coprime (their HCF is 1, i.e., \(\text{gcd}(x, y) = 1\)). In this case, the LCM is \(L = Hxy\).
Analyzing the Given Information
We are given:
- The ratio of LCM to HCF is \(3:2\). This means \(\frac{L}{H} = \frac{3}{2}\).
- The sum of the two numbers is 45. That is, \(a + b = 45\).
We need to find the product of the numbers, \(a \times b\).
Checking for Consistency Using HCF and LCM Properties
From the property \(L = Hxy\), we know that \(\frac{L}{H} = xy\). Here, \(x\) and \(y\) are the coprime factors when the numbers are expressed in terms of their HCF (\(a=Hx, b=Hy\)). Since \(a\) and \(b\) are integers, and \(H\) must also be an integer (as it's the HCF of integers), \(x = \frac{a}{H}\) and \(y = \frac{b}{H}\) must also be integers. Furthermore, \(x\) and \(y\) must be coprime integers.
Based on the given ratio, \(\frac{L}{H} = \frac{3}{2}\).
So, we have \(xy = \frac{3}{2}\).
However, as we established, \(x\) and \(y\) must be integers because they are coprime factors relative to the HCF of integers. The product of two integers (\(x \times y\)) must always be an integer.
The value \(\frac{3}{2}\) is not an integer.
This creates a contradiction: the product \(xy\) must be an integer, but the given ratio \(\frac{L}{H} = \frac{3}{2}\) implies \(xy\) is not an integer.
Conclusion Based on the Contradiction
Since the relationship \(\frac{L}{H} = \frac{3}{2}\) violates a fundamental property that the ratio of LCM to HCF for two integers must be an integer, it means that no pair of positive integers can exist that satisfies the given condition \(L:H = 3:2\).
Because no such numbers \(a\) and \(b\) can exist under the given conditions, it is impossible to determine their product.
Therefore, the product of the numbers cannot be determined due to insufficient or inconsistent data provided in the problem statement.
Revision Table: HCF and LCM Properties
| Property | Description |
|---|---|
| Product Relation | Product of two numbers = HCF × LCM |
| Divisibility | HCF always divides LCM exactly (L is a multiple of H) |
| Relative Primes | If \(a=Hx, b=Hy\) where \(\text{gcd}(x,y)=1\), then \(L=Hxy\) |
| Ratio L/H | \(\frac{L}{H} = xy\), where \(x, y\) are coprime integers. This ratio must be an integer. |
Additional Information: Why L/H Must Be Integer
Let the two numbers be \(a\) and \(b\). Let their HCF be \(H\). We can write \(a = Hx\) and \(b = Hy\), where \(x\) and \(y\) are integers that have no common factors other than 1 (i.e., \(\text{gcd}(x, y) = 1\)).
The LCM of \(a\) and \(b\) is found by taking the HCF and multiplying by the remaining coprime factors. So, \(L = H \times x \times y\).
Now, let's look at the ratio \(\frac{L}{H}\).
\(\frac{L}{H} = \frac{Hxy}{H} = xy\)
Since \(x\) and \(y\) are integers (as they are obtained by dividing integers \(a\) and \(b\) by their integer HCF \(H\)), their product \(xy\) must also be an integer.
Thus, for any pair of positive integers, the ratio of their LCM to their HCF must always be an integer.
The given condition that the ratio \(L:H\) is \(3:2\), meaning \(\frac{L}{H} = \frac{3}{2}\), directly contradicts this fundamental property. Therefore, no such pair of numbers exists, and the product cannot be determined.
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