Suppose p and q are the LCM and HCF respectively of two positive numbers. If p : q = 14 : 1 and pq = 1134, then what is the difference between the two numbers?
45
Understanding the Problem: Finding the Difference of Two Numbers
The question provides information about the Least Common Multiple (LCM) and Highest Common Factor (HCF) of two positive numbers. Let the two positive numbers be \(a\) and \(b\). We are given that their LCM is \(p\) and their HCF is \(q\). We are provided with two conditions:
- The ratio of \(p\) to \(q\) is \(14 : 1\), which can be written as \(p/q = 14/1\) or \(p = 14q\).
- The product of \(p\) and \(q\) is \(1134\), i.e., \(pq = 1134\).
Our goal is to find the absolute difference between the two numbers, which is \(|a - b|\).
Relating LCM, HCF, and the Numbers
A fundamental property connecting two positive numbers, their LCM, and their HCF is:
\(a \times b = \text{LCM}(a, b) \times \text{HCF}(a, b)\)
In our case, this means:
\(a \times b = p \times q\)
We are given \(pq = 1134\), so the product of the two numbers is \(a \times b = 1134\).
Finding the Values of LCM (p) and HCF (q)
We have the following system of equations based on the given information:
- \(p = 14q\)
- \(pq = 1134\)
We can substitute the first equation into the second one to solve for \(q\):
\((14q) \times q = 1134\)
\(14q^2 = 1134\)
Now, we solve for \(q^2\):
\(q^2 = \frac{1134}{14}\)
\(q^2 = 81\)
Since \(q\) is the HCF of positive numbers, it must be positive. Therefore, we take the positive square root:
\(q = \sqrt{81} = 9\)
So, the HCF is \(q = 9\).
Now we can find \(p\) using the relation \(p = 14q\):
\(p = 14 \times 9\)
\(p = 126\)
So, the LCM is \(p = 126\).
Finding the Two Numbers
We know that the HCF of the two numbers \(a\) and \(b\) is \(q = 9\). This means that \(a\) and \(b\) can be expressed as multiples of 9. Let:
\(a = 9x\)
\(b = 9y\)
where \(x\) and \(y\) are positive integers and they must be coprime (their HCF is 1). If \(x\) and \(y\) had a common factor greater than 1, then the HCF of \(a\) and \(b\) would be greater than 9.
We also know that \(a \times b = 1134\). Substituting \(a = 9x\) and \(b = 9y\):
\((9x) \times (9y) = 1134\)
\(81xy = 1134\)
Now, we solve for \(xy\):
\(xy = \frac{1134}{81}\)
\(xy = 14\)
We need to find pairs of positive integers \((x, y)\) such that their product is 14 AND their HCF is 1 (coprime). The pairs of factors for 14 are:
- (1, 14)
- (2, 7)
- (7, 2)
- (14, 1)
Let's check the HCF for each pair:
- HCF(1, 14) = 1. This pair is coprime.
- HCF(2, 7) = 1. This pair is coprime.
- HCF(7, 2) = 1. This pair is coprime.
- HCF(14, 1) = 1. This pair is coprime.
All four pairs are coprime. Now we find the corresponding numbers \((a, b)\) using \(a = 9x\) and \(b = 9y\):
- Using (x, y) = (1, 14): \(a = 9 \times 1 = 9\), \(b = 9 \times 14 = 126\). The numbers are 9 and 126.
- Using (x, y) = (2, 7): \(a = 9 \times 2 = 18\), \(b = 9 \times 7 = 63\). The numbers are 18 and 63.
- Using (x, y) = (7, 2): \(a = 9 \times 7 = 63\), \(b = 9 \times 2 = 18\). The numbers are 63 and 18.
- Using (x, y) = (14, 1): \(a = 9 \times 14 = 126\), \(b = 9 \times 1 = 9\). The numbers are 126 and 9.
The possible pairs of numbers are (9, 126) and (18, 63).
Calculating the Difference Between the Two Numbers
We need to find the difference between the two numbers for each pair:
- For the pair (9, 126), the difference is \(|126 - 9| = 117\).
- For the pair (18, 63), the difference is \(|63 - 18| = 45\).
We check the options provided in the question:
| Option | Difference |
|---|---|
| 1 | 27 |
| 2 | 35 |
| 3 | 45 |
| 4 | cannot be determined due to insufficient data |
Comparing the calculated differences (117 and 45) with the options, we see that 45 is one of the options (Option 3). The difference 117 is not an option.
Therefore, the two numbers must be 18 and 63, and their difference is 45.
Conclusion
By using the properties of LCM and HCF, the given ratio, and the product, we were able to determine the possible pairs of numbers and calculate their differences. The difference that matches one of the given options is 45.
Revision Table: Key Steps for LCM and HCF Problems
| Step | Action | Details |
|---|---|---|
| 1 | Define variables | Let numbers be \(a, b\), LCM be \(p\), HCF be \(q\). |
| 2 | Note given conditions | Write down ratios, products, etc. (e.g., \(p:q = 14:1\), \(pq = 1134\)). |
| 3 | Use \(ab = pq\) property | Connect the product of numbers to LCM and HCF. |
| 4 | Solve for \(p\) and \(q\) | Use the given conditions to find the values of LCM and HCF. |
| 5 | Express numbers using HCF | Set \(a = qx\) and \(b = qy\) where \(x, y\) are coprime integers. |
| 6 | Use \(ab = pq\) or \(p = qxy\) | Find the value of \(xy\). |
| 7 | Find coprime pairs \((x, y)\) | List pairs whose product is \(xy\) and HCF is 1. |
| 8 | Calculate possible \((a, b)\) pairs | Substitute \((x, y)\) back into \(a=qx, b=qy\). |
| 9 | Calculate the required value | Find the difference, sum, etc., asked in the question for the pairs. |
| 10 | Match with options | Compare results with the given options to find the answer. |
Additional Information on LCM and HCF
- Definition of HCF: The Highest Common Factor (HCF) or Greatest Common Divisor (GCD) of two or more positive integers is the largest positive integer that divides all the numbers without leaving a remainder.
- Definition of LCM: The Least Common Multiple (LCM) of two or more positive integers is the smallest positive integer that is a multiple of all the numbers.
- Relationship \(ab = pq\): This property \(a \times b = \text{LCM}(a, b) \times \text{HCF}(a, b)\) is very useful for problems involving two numbers. It holds true for any two positive integers \(a\) and \(b\).
- Representing numbers using HCF: If the HCF of two numbers \(a\) and \(b\) is \(q\), then \(a\) and \(b\) can always be written as \(a = qx\) and \(b = qy\), where \(x\) and \(y\) are integers that are coprime (HCF(x, y) = 1). This representation is crucial for finding the numbers when the HCF is known.
Similar Questions
If (x + k) is the HCF of x 2+ 5x + 6 and x 2+ 8x + 15, then what is the value of k?
A floor of a big hall has dimensions 30 m 60 cm and 23 m 40 cm. It is to be paved with square tiles of same size. What is the minimum number of tiles required ?
The LCM of two prime numbers p and q is 2231, where p > q. What is the value of p - q ?
Three runners are running in a circular track, and they complete one round in 20, 30 and 35 minutes respectively. When will they next meet at the starting point ?
What is the least perfect square which is divisible by 3, 4, 5, 6 and 7?
If (x - k) is the HCF of x 2+ ax + b and x 2+ cx + d, then what is the value of k?
HCF of two numbers is 12. Which one of the following can never be their LCM?
X, Y and Z start at some point and same time in the same direction to run around a circular stadium. X completes a round in 252 seconds, Y in 308 seconds and Z in 198 seconds. After what time will they meet again at the starting point?
What is the LCM of the polynomials x 3+ 3x 2+ 3x + 1, x 3+ 5x 2+ 5x + 4 and x 2+ 5x + 4?
HCF and LCM of two polynomials are (x + 3) and (x 3- 9x 2- x + 105). If one of the two polynomials is (x 2- 4x - 21), then the other is
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?
