Consider the equation, 1/x +1/y = 1/z, where x,y,z are natural numbers.
If x = 12, find the total number of solutions to this equation.

Options

A. 6
B. 1
C. 18
D. 9

Answer

To solve the equation \( \frac{1}{x} + \frac{1}{y} = \frac{1}{z} \) where \( x, y, z \) are natural numbers and \( x = 12 \), we can follow these steps:

### Step 1: Substitute the value of \( x \)
We start by substituting \( x = 12 \) into the equation:
\[ \frac{1}{12} + \frac{1}{y} = \frac{1}{z} \]

### Step 2: Rearrange the equation
Next, we rearrange the equation to isolate \( \frac{1}{y} \):
\[ \frac{1}{y} = \frac{1}{z} - \frac{1}{12} \]
This can be rewritten as:
\[ \frac{1}{y} = \frac{12 - z}{12z} \]

### Step 3: Take the reciprocal
Taking the reciprocal gives us:
\[ y = \frac{12z}{12 - z} \]

### Step 4: Ensure \( y \) is a natural number
For \( y \) to be a natural number, \( 12z \) must be divisible by \( 12 - z \). This means \( 12 - z \) must be a divisor of \( 12z \).

### Step 5: Find the divisors of \( 12z \)
We can express \( 12z \) in terms of its divisors. The possible values for \( z \) must satisfy \( 12 - z > 0 \), which implies \( z < 12 \).

### Step 6: List possible values for \( z \)
The possible values for \( z \) are \( 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 \).

### Step 7: Check each value of \( z \)
We will check each value of \( z \) from \( 1 \) to \( 11 \) to see if \( y \) is a natural number:
- For \( z = 1 \): \( y = \frac{12 \cdot 1}{12 - 1} = \frac{12}{11} \) (not natural)
- For \( z = 2 \): \( y = \frac{12 \cdot 2}{12 - 2} = \frac{24}{10} = 2.4 \) (not natural)
- For \( z = 3 \): \( y = \frac{12 \cdot 3}{12 - 3} = \frac{36}{9} = 4 \) (natural)
- For \( z = 4 \): \( y = \frac{12 \cdot 4}{12 - 4} = \frac{48}{8} = 6 \) (natural)
- For \( z = 5 \): \( y = \frac{12 \cdot 5}{12 - 5} = \frac{60}{7} \) (not natural)
- For \( z = 6 \): \( y = \frac{12 \cdot 6}{12 - 6} = \frac{72}{6} = 12 \) (natural)
- For \( z = 7 \): \( y = \frac{12 \cdot 7}{12 - 7} = \frac{84}{5} \) (not natural)
- For \( z = 8 \): \( y = \frac{12 \cdot 8}{12 - 8} = \frac{96}{4} = 24 \) (natural)
- For \( z = 9 \): \( y = \frac{12 \cdot 9}{12 - 9} = \frac{108}{3} = 36 \) (natural)
- For \( z = 10 \): \( y = \frac{12 \cdot 10}{12 - 10} = \frac{120}{2} = 60 \) (natural)
- For \( z = 11 \): \( y = \frac{12 \cdot 11}{12 - 11} = \frac{132}{1} = 132 \) (natural)

### Step 8: Count the valid solutions
The valid pairs \((y, z)\) that yield natural numbers are:
1. \( (4, 3) \)
2. \( (6, 4) \)
3. \( (12, 6) \)
4. \( (24, 8) \)
5. \( (36, 9) \)
6. \( (60, 10) \)
7. \( (132, 11) \)

Thus, we have a total of **7 solutions**.

### Final Answer
The total number of solutions to the equation \( \frac{1}{x} + \frac{1}{y} = \frac{1}{z} \) when \( x = 12 \) is **7**.

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