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:

1. \(p = 14q\)
2. \(pq = 1134\)

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:

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

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.