The number of positive integer solutions of a+b+c=60, where a is a factor of b and c, is

Options

A. 184
B. 200
C. 144
D. 270

Explanation

To solve the problem of finding the number of positive integer solutions for the equation \( a + b + c = 60 \) where \( a \) is a factor of both \( b \) and \( c \), we can follow these steps:

### Step 1: Express \( b \) and \( c \) in terms of \( a \)
Since \( a \) is a factor of both \( b \) and \( c \), we can express them as:
\[ b = x \cdot a \quad \text{and} \quad c = y \cdot a \]
where \( x \) and \( y \) are positive integers.

### Step 2: Substitute into the equation
Substituting \( b \) and \( c \) into the original equation gives:
\[ a + x \cdot a + y \cdot a = 60 \]
This simplifies to:
\[ a(1 + x + y) = 60 \]

### Step 3: Rearrange the equation
From the above equation, we can express \( 1 + x + y \) as:
\[ 1 + x + y = \frac{60}{a} \]
Thus, we have:
\[ x + y = \frac{60}{a} - 1 \]

### Step 4: Determine the possible values for \( a \)
Since \( a \) must be a positive integer that divides 60, we need to find the divisors of 60. The positive divisors of 60 are:
\[ 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 \]

### Step 5: Calculate the number of solutions for each divisor
For each divisor \( a \), we will calculate \( x + y \) and then find the number of positive integer solutions for \( x + y \).

1. **For \( a = 1 \)**:
\[ x + y = \frac{60}{1} - 1 = 59 \quad \Rightarrow \quad \text{Number of solutions} = 58 \]

2. **For \( a = 2 \)**:
\[ x + y = \frac{60}{2} - 1 = 29 \quad \Rightarrow \quad \text{Number of solutions} = 28 \]

3. **For \( a = 3 \)**:
\[ x + y = \frac{60}{3} - 1 = 19 \quad \Rightarrow \quad \text{Number of solutions} = 18 \]

4. **For \( a = 4 \)**:
\[ x + y = \frac{60}{4} - 1 = 14 \quad \Rightarrow \quad \text{Number of solutions} = 13 \]

5. **For \( a = 5 \)**:
\[ x + y = \frac{60}{5} - 1 = 11 \quad \Rightarrow \quad \text{Number of solutions} = 10 \]

6. **For \( a = 6 \)**:
\[ x + y = \frac{60}{6} - 1 = 9 \quad \Rightarrow \quad \text{Number of solutions} = 8 \]

7. **For \( a = 10 \)**:
\[ x + y = \frac{60}{10} - 1 = 5 \quad \Rightarrow \quad \text{Number of solutions} = 4 \]

8. **For \( a = 12 \)**:
\[ x + y = \frac{60}{12} - 1 = 4 \quad \Rightarrow \quad \text{Number of solutions} = 3 \]

9. **For \( a = 15 \)**:
\[ x + y = \frac{60}{15} - 1 = 3 \quad \Rightarrow \quad \text{Number of solutions} = 2 \]

10. **For \( a = 20 \)**:
\[ x + y = \frac{60}{20} - 1 = 2 \quad \Rightarrow \quad \text{Number of solutions} = 1 \]

11. **For \( a = 30 \)**:
\[ x + y = \frac{60}{30} - 1 = 1 \quad \Rightarrow \quad \text{Number of solutions} = 0 \quad (\text{not valid since } x, y \text{ must be positive}) \]

12. **For \( a = 60 \)**:
\[ x + y = \frac{60}{60} - 1 = 0 \quad \Rightarrow \quad \text{Number of solutions} = 0 \quad (\text{not valid since } x, y \text{ must be positive}) \]

### Step 6: Sum the number of solutions
Now we sum the valid solutions:
\[ 58 + 28 + 18 + 13 + 10 + 8 + 4 + 3 + 2 + 1 = 144 \]

### Final Answer
The total number of positive integer solutions for \( a + b + c = 60 \) where \( a \) is a factor of both \( b \) and \( c \) is **144**.

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