There exist positive integers A, B and C with no common factors greater than 1, such that 1, such that `A log_(200) 5+ B log_(200) 2=C` The sum `A + B+ C` equals

Options

A. 5
B. 6
C. 7
D. 8

Explanation

To solve the equation \( A \log_{200} 5 + B \log_{200} 2 = C \) for positive integers \( A, B, \) and \( C \) with no common factors greater than 1, we can follow these steps:

### Step 1: Use the Change of Base Formula
We start by applying the change of base formula for logarithms, which states:
\[ \log_{x} y = \frac{\log y}{\log x} \]
Using this, we can rewrite the equation:
\[ A \log_{200} 5 + B \log_{200} 2 = C \implies A \frac{\log 5}{\log 200} + B \frac{\log 2}{\log 200} = C \]
Multiplying through by \( \log 200 \) gives:
\[ A \log 5 + B \log 2 = C \log 200 \]

### Step 2: Express \( \log 200 \)
Next, we can express \( \log 200 \) in terms of \( \log 5 \) and \( \log 2 \):
\[ 200 = 2^3 \times 5^2 \implies \log 200 = \log(2^3) + \log(5^2) = 3 \log 2 + 2 \log 5 \]
Substituting this back into our equation gives:
\[ A \log 5 + B \log 2 = C (3 \log 2 + 2 \log 5) \]

### Step 3: Rearranging the Equation
Rearranging the equation, we can group the terms:
\[ A \log 5 + B \log 2 = 3C \log 2 + 2C \log 5 \]
This leads to:
\[ (A - 2C) \log 5 + (B - 3C) \log 2 = 0 \]

### Step 4: Setting Coefficients to Zero
Since \( \log 5 \) and \( \log 2 \) are independent, we can set the coefficients of each logarithm to zero:
1. \( A - 2C = 0 \)
2. \( B - 3C = 0 \)

From these equations, we can express \( A \) and \( B \) in terms of \( C \):
\[ A = 2C \quad \text{and} \quad B = 3C \]

### Step 5: Finding Values for \( A, B, \) and \( C \)
To ensure that \( A, B, \) and \( C \) have no common factors greater than 1, we can choose \( C = 1 \):
\[ A = 2 \cdot 1 = 2, \quad B = 3 \cdot 1 = 3, \quad C = 1 \]

### Step 6: Calculate the Sum \( A + B + C \)
Now we can find the sum:
\[ A + B + C = 2 + 3 + 1 = 6 \]

Thus, the final answer is:
\[ \boxed{6} \]