The sum of the cubes of two numbers is 793. The sum of the numbers is 13. Then the difference of the two numbers is

Options

A. a) 7
B. b) 6
C. c) 5
D. d) 8

Answer

To solve the problem step by step, we will denote the two numbers as \( A \) and \( B \).

### Step 1: Set up the equations
We are given two conditions:
1. The sum of the cubes of the two numbers:
\[ A^3 + B^3 = 793 \]
2. The sum of the two numbers:
\[ A + B = 13 \]

### Step 2: Use the identity for the sum of cubes
We can use the identity for the sum of cubes:
\[ A^3 + B^3 = (A + B)(A^2 - AB + B^2) \]
We can also express \( A^2 + B^2 \) in terms of \( A + B \) and \( AB \):
\[ A^2 + B^2 = (A + B)^2 - 2AB \]
Thus, we can rewrite the sum of cubes as:
\[ A^3 + B^3 = (A + B)((A + B)^2 - 3AB) \]

### Step 3: Substitute the known values
Substituting \( A + B = 13 \) into the equation:
\[ A^3 + B^3 = 13((13)^2 - 3AB) \]
Calculating \( (13)^2 \):
\[ (13)^2 = 169 \]
So we have:
\[ A^3 + B^3 = 13(169 - 3AB) \]
Setting this equal to 793:
\[ 13(169 - 3AB) = 793 \]

### Step 4: Solve for \( AB \)
Dividing both sides by 13:
\[ 169 - 3AB = \frac{793}{13} \]
Calculating \( \frac{793}{13} \):
\[ \frac{793}{13} = 61 \]
Now we have:
\[ 169 - 3AB = 61 \]
Rearranging gives:
\[ 3AB = 169 - 61 \]
Calculating \( 169 - 61 \):
\[ 3AB = 108 \]
Dividing by 3:
\[ AB = \frac{108}{3} = 36 \]

### Step 5: Use the values of \( A + B \) and \( AB \) to find \( A - B \)
We know:
- \( A + B = 13 \)
- \( AB = 36 \)

Using the identity for the square of the difference:
\[ (A - B)^2 = (A + B)^2 - 4AB \]
Substituting the known values:
\[ (A - B)^2 = 13^2 - 4 \times 36 \]
Calculating \( 13^2 \):
\[ 13^2 = 169 \]
Calculating \( 4 \times 36 \):
\[ 4 \times 36 = 144 \]
Now substituting these values:
\[ (A - B)^2 = 169 - 144 = 25 \]

### Step 6: Take the square root to find \( A - B \)
Taking the square root:
\[ A - B = \sqrt{25} = 5 \]

### Final Answer
The difference of the two numbers \( A - B \) is \( 5 \).

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