Find the value of abc, if a, b and c are integers amd `a+1/(b+1/c) = (37)/(16)`

Options

A. 30
B. 24
C. 48
D. none

Answer

To solve the equation \( a + \frac{1}{b + \frac{1}{c}} = \frac{37}{16} \) for integers \( a \), \( b \), and \( c \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ a + \frac{1}{b + \frac{1}{c}} = \frac{37}{16} \]
We can express \( \frac{1}{b + \frac{1}{c}} \) in a simpler form.

### Step 2: Simplify the fraction
The term \( \frac{1}{b + \frac{1}{c}} \) can be rewritten as:
\[ \frac{1}{b + \frac{1}{c}} = \frac{1}{\frac{bc + 1}{c}} = \frac{c}{bc + 1} \]
Thus, we can rewrite the original equation as:
\[ a + \frac{c}{bc + 1} = \frac{37}{16} \]

### Step 3: Isolate \( \frac{c}{bc + 1} \)
Rearranging gives us:
\[ \frac{c}{bc + 1} = \frac{37}{16} - a \]

### Step 4: Find a suitable integer \( a \)
Since \( a \) is an integer, we can try different integer values for \( a \) to see which one allows \( \frac{37}{16} - a \) to be a fraction that can be expressed as \( \frac{c}{bc + 1} \).

Trying \( a = 2 \):
\[ \frac{37}{16} - 2 = \frac{37 - 32}{16} = \frac{5}{16} \]

### Step 5: Set up the equation
Now we have:
\[ \frac{c}{bc + 1} = \frac{5}{16} \]

### Step 6: Cross-multiply
Cross-multiplying gives:
\[ 16c = 5(bc + 1) \]
Expanding this, we have:
\[ 16c = 5bc + 5 \]

### Step 7: Rearranging the equation
Rearranging gives:
\[ 5bc - 16c + 5 = 0 \]

### Step 8: Factor the equation
Factoring out \( c \):
\[ c(5b - 16) = -5 \]
This means \( c \) must be a divisor of \(-5\). The integer divisors of \(-5\) are \(-5, -1, 1, 5\).

### Step 9: Test values for \( c \)
1. If \( c = 1 \):
\[ 5b - 16 = -5 \implies 5b = 11 \implies b = \frac{11}{5} \text{ (not an integer)} \]
2. If \( c = -1 \):
\[ 5b - 16 = 5 \implies 5b = 21 \implies b = \frac{21}{5} \text{ (not an integer)} \]
3. If \( c = 5 \):
\[ 5b - 16 = -1 \implies 5b = 15 \implies b = 3 \text{ (integer)} \]
4. If \( c = -5 \):
\[ 5b - 16 = 1 \implies 5b = 17 \implies b = \frac{17}{5} \text{ (not an integer)} \]

### Step 10: Final values
From our tests, we find:
- \( a = 2 \)
- \( b = 3 \)
- \( c = 5 \)

### Step 11: Calculate \( abc \)
Now we calculate:
\[ abc = 2 \times 3 \times 5 = 30 \]

Thus, the value of \( abc \) is \( \boxed{30} \).