A certain number 'C' when divided by `N_1` it leaves a remainder of 13 and when it is divided by `N_2` it leaves a remainder of 1, where `N_1` and `N_2` are the positive integers. Then the value of `N_1 + N_2` is if `(N_1)/(N_2) = 5/4`.

Options

A. 36
B. 27
C. 54
D. can't be determined uniquely

Answer

To solve the problem step by step, we will use the information provided in the question and the relationships between the numbers.

### Step 1: Set Up the Equations
Given:
1. \( C \equiv 13 \mod N_1 \) (C leaves a remainder of 13 when divided by \( N_1 \))
2. \( C \equiv 1 \mod N_2 \) (C leaves a remainder of 1 when divided by \( N_2 \))
3. The ratio \( \frac{N_1}{N_2} = \frac{5}{4} \)

From the first equation, we can express \( C \) as:
\[ C = kN_1 + 13 \]
for some integer \( k \).

From the second equation, we can express \( C \) as:
\[ C = mN_2 + 1 \]
for some integer \( m \).

### Step 2: Equate the Two Expressions for C
Since both expressions equal \( C \), we can set them equal to each other:
\[ kN_1 + 13 = mN_2 + 1 \]

Rearranging gives us:
\[ kN_1 - mN_2 = -12 \]
or
\[ kN_1 = mN_2 - 12 \]

### Step 3: Express N1 in Terms of N2
From the ratio \( \frac{N_1}{N_2} = \frac{5}{4} \), we can express \( N_1 \) in terms of \( N_2 \):
\[ N_1 = \frac{5}{4} N_2 \]

### Step 4: Substitute N1 into the Equation
Substituting \( N_1 \) into the equation \( kN_1 - mN_2 = -12 \):
\[ k\left(\frac{5}{4}N_2\right) - mN_2 = -12 \]

This simplifies to:
\[ \frac{5k}{4}N_2 - mN_2 = -12 \]

Factoring out \( N_2 \):
\[ N_2\left(\frac{5k}{4} - m\right) = -12 \]

### Step 5: Solve for N2
From this equation, we can express \( N_2 \) as:
\[ N_2 = \frac{-12}{\frac{5k}{4} - m} \]

### Step 6: Find N1 and N2
Since \( N_2 \) must be a positive integer, \( \frac{5k}{4} - m \) must be a negative integer, which means \( m \) must be greater than \( \frac{5k}{4} \).

Now, let's express \( N_1 \) in terms of \( k \):
\[ N_1 = \frac{5}{4}N_2 = \frac{5}{4} \cdot \frac{-12}{\frac{5k}{4} - m} \]

### Step 7: Calculate N1 + N2
Now we can find \( N_1 + N_2 \):
\[ N_1 + N_2 = N_2\left(\frac{5}{4} + 1\right) = N_2 \cdot \frac{9}{4} \]

Substituting \( N_2 \):
\[ N_1 + N_2 = \frac{-12}{\frac{5k}{4} - m} \cdot \frac{9}{4} \]

### Step 8: Determine Values
To find specific integer values for \( N_1 \) and \( N_2 \), we can try different integer values for \( k \) and \( m \) that satisfy the conditions.

### Final Step: Conclusion
After testing various integers, we can find that:
- If \( N_1 = 20 \) and \( N_2 = 16 \), then \( N_1 + N_2 = 36 \).
- Therefore, the answer is \( N_1 + N_2 = 36 \).