If LCM(27, n) = 54, and HCF(27, n) = 9, then the value of n is:
18
Finding the Value of n using LCM and HCF
This problem asks us to find the value of an unknown number, represented by 'n', given the Least Common Multiple (LCM) and Highest Common Factor (HCF) of 27 and 'n'. We are provided with the following information:
- The first number is 27.
- The LCM of 27 and 'n' is 54, i.e., \(\text{LCM}(27, n) = 54\).
- The HCF of 27 and 'n' is 9, i.e., \(\text{HCF}(27, n) = 9\).
Understanding the LCM and HCF Relationship
There's a fundamental property that connects two numbers (let's call them 'a' and 'b') with their LCM and HCF. This property states that the product of the two numbers is equal to the product of their LCM and HCF.
Mathematically, this is represented as:
\(a \times b = \text{HCF}(a, b) \times \text{LCM}(a, b)\)
Applying the Formula to Find n
In this specific problem, we have:
- \(a = 27\)
- \(b = n\)
- \(\text{HCF}(27, n) = 9\)
- \(\text{LCM}(27, n) = 54\)
Let's substitute these values into the formula:
\(27 \times n = \text{HCF}(27, n) \times \text{LCM}(27, n)\)
\(27 \times n = 9 \times 54\)
Calculating the Value of n
First, calculate the product of the HCF and LCM:
\(9 \times 54 = 486\)
Now, our equation becomes:
\(27 \times n = 486\)
To find 'n', we need to divide 486 by 27:
\(n = \frac{486}{27}\)
Performing the division:
\(n = 18\)
Verification of the Result
Let's check if \(n = 18\) satisfies the given conditions:
- Check HCF(27, 18): The factors of 27 are 1, 3, 9, 27. The factors of 18 are 1, 2, 3, 6, 9, 18. The highest common factor is indeed 9. So, \(\text{HCF}(27, 18) = 9\). This matches the given information.
- Check LCM(27, 18): The multiples of 27 are 27, 54, 81, ... The multiples of 18 are 18, 36, 54, 72, ... The least common multiple is 54. So, \(\text{LCM}(27, 18) = 54\). This also matches the given information.
Since both conditions are met, the calculated value of \(n = 18\) is correct.
Similar Questions
- Three bells ring at intervals of 15, 30 and 45 minutes respectively. At what time will they ring together again, if they rang simultaneously at 8.00 AM?
Important Questions from LCM and HCF
The greatest three-digit number which is divisible by 14, 28, and 42 is:
What is the greatest number that will divide 209 and 347 leaving remainder 5 and 7 respectively?
A number is three times another number and their HCF is 8. What is the sum of the squares of the numbers?
The HCF of 2091, 3485 and 4879 is x. The sum of the digits of x is:
If three numbers are in ratio of 3 : 5 : 7 and their LCM is 2415, what is the difference between the second number and the first number?
