Question : LCM of three numbers \(p, q \text{ and } r \, (p > q > r > 0)\) is 30 and their HCF is 5. What is the product of the three numbers ?
Statement-I : \(p = 15\)
Statement-II : \(q = 10\)
Which one of the following is correct in respect of the above Question and the Statements ?
The Question can be answered by using one of the statements alone, but cannot be answered using the other statement alone
To solve the given question, we need to find out the product of three numbers \( p, q, \text{ and } r \) whose LCM is 30 and HCF is 5, using the provided statements.
Given:
- \(\text{LCM}(p, q, r) = 30\)
- \(\text{HCF}(p, q, r) = 5\)
The relationship between the LCM, HCF, and the product of the numbers is given by the formula:
\(p \times q \times r = \text{LCM}(p, q, r) \times \text{HCF}(p, q, r)\)
Substituting the known values, we have:
\(p \times q \times r = 30 \times 5 = 150\)
Now, let us analyze the given statements:
- Statement I: \( p = 15 \)
- Statement II: \( q = 10 \)
Let's check if either statement alone or both together can determine the product of the three numbers:
Using Statement I alone:
We only know that \( p = 15 \). To find the other two numbers \( q \) and \( r \), we need additional information, as we cannot determine them uniquely from this statement alone.
Using Statement II alone:
Similarly, with \( q = 10 \), we cannot determine \( p \) and \( r \) uniquely without additional information. Therefore, Statement II alone is insufficient.
Using both Statements I and II together:
If \( p = 15 \) and \( q = 10 \), we can find the third number \( r \) using:
\(p \times q \times r = 150 \rightarrow 15 \times 10 \times r = 150\)
Simplifying, we get:
\(150 \times r = 150 \rightarrow r = \frac{150}{150} = 1\)
Thus, the numbers are \( p = 15 \), \( q = 10 \), and \( r = 1 \). Their product is 150, satisfying the condition:
\(15 \times 10 \times 1 = 150\), and the calculated LCM is 30, which matches the given LCM.
Therefore, the correct answer is that the question can be answered by using both the statements together, but cannot be answered using either statement alone.
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Important Questions from LCM and HCF
The HCF and LCM of two numbers are 12 and 72, respectively. If the ratio of the two numbers is 2 ∶ 3, then the larger of the two numbers is:
Find the greatest number that will divide 43, 91 and 183 so as to leave the same remainder in each case.
Joseph visits the club on every 5 th day, Harsh visits on every 24 th day, while Sumit visits on every 9 th day. If all three of them met at the club on a Sunday, then on which day will all three of them meet again?
What is the least number which when divided by 12,20 and 24 leaves in each case a remainder of 8?
Which of the following is a pair of co-primes?
