Understanding the Problem: Finding the Other Number

The question provides information about the Least Common Multiple (LCM) and Highest Common Factor (HCF) of two numbers. We are given the sum and the difference of their LCM and HCF. Additionally, one of the two numbers is provided. Our goal is to find the value of the other number.

Setting up Equations for LCM and HCF

Let L represent the LCM and H represent the HCF of the two numbers. Based on the problem statement, we have two equations:

1. The sum of the LCM and HCF is 512:
   \(L + H = 512\)
2. The difference between the LCM and HCF is 496:
   \(L - H = 496\)

We can solve these two linear equations simultaneously to find the values of L and H.

Solving for LCM and HCF

Adding the two equations:

\((L + H) + (L - H) = 512 + 496\)

\(2L = 1008\)

\(L = \frac{1008}{2}\)

\(L = 504\)

So, the LCM of the two numbers is 504.

Now, substitute the value of L into the first equation (\(L + H = 512\)):

\(504 + H = 512\)

\(H = 512 - 504\)

\(H = 8\)

Thus, the HCF of the two numbers is 8.

Relationship between Numbers, LCM, and HCF

A fundamental property relating two positive integers, their LCM, and their HCF is that the product of the two numbers is equal to the product of their LCM and HCF.

If the two numbers are N1 and N2, then:

\(N1 \times N2 = \text{LCM} \times \text{HCF}\)

Finding the Other Number

We are given that one number (N1) is 72. We have calculated the LCM as 504 and the HCF as 8. Let the other number be N2.

Using the relationship:

\(72 \times N2 = 504 \times 8\)

Now, we can solve for N2:

\(N2 = \frac{504 \times 8}{72}\)

To simplify the calculation, we can notice that 72 is a multiple of 8 (\(72 = 8 \times 9\)).

\(N2 = \frac{504 \times 8}{9 \times 8}\)

Cancel out the common factor of 8:

\(N2 = \frac{504}{9}\)

Perform the division:

\(504 \div 9\)

We can do this by dividing 504 by 9:

50 divided by 9 is 5 with a remainder of 5 (9 * 5 = 45, 50 - 45 = 5).

Bring down the 4, making the new number 54.

54 divided by 9 is 6 (9 * 6 = 54).

So, \(504 \div 9 = 56\).

\(N2 = 56\)

The other number is 56.

Conclusion

By using the given sum and difference to find the LCM and HCF, and then applying the property that the product of two numbers equals the product of their LCM and HCF, we found the other number.

The other number is 56.

Revision Table: Key Concepts

Additional Information: Solving Systems of Equations

In this problem, we used a simple system of two linear equations to find the values of LCM and HCF. The method used was elimination, which is useful when the coefficients of one variable are opposites (like +H and -H here) or the same.

Consider the general form of such a system:

\(x + y = a\)

\(x - y = b\)

Adding the equations:

\((x + y) + (x - y) = a + b\)

\(2x = a + b\)

\(x = \frac{a + b}{2}\)

Substituting x into the first equation:

\(\frac{a + b}{2} + y = a\)

\(y = a - \frac{a + b}{2}\)

\(y = \frac{2a - (a + b)}{2}\)

\(y = \frac{2a - a - b}{2}\)

\(y = \frac{a - b}{2}\)

So, if you know the sum (a) and difference (b) of two quantities (x and y), the quantities can be found as half of their sum and half of their difference added/subtracted from the sum/average, or more simply, x = (a+b)/2 and y = (a-b)/2. In our case, L and H are the quantities, 512 is the sum (a), and 496 is the difference (b).

• \(L = \frac{512 + 496}{2} = \frac{1008}{2} = 504\)
• \(H = \frac{512 - 496}{2} = \frac{16}{2} = 8\)

This confirms our earlier calculation for LCM and HCF.