Which of the following numbers is not a factor of `5^(p)7^(q)` (`p ne 0`, `q ne 0`)

Options

A. 35
B. 175
C. 1225
D. 735

Answer

To determine which of the following numbers is not a factor of \(5^p 7^q\) (where \(p \neq 0\) and \(q \neq 0\)), we need to analyze each option given in the question. Let's break down the steps:

### Step 1: Understand the expression \(5^p 7^q\)
The expression \(5^p 7^q\) indicates that the number has factors of \(5\) raised to the power of \(p\) and \(7\) raised to the power of \(q\). Since \(p\) and \(q\) are both non-zero, the number must include at least one factor of \(5\) and one factor of \(7\).

### Step 2: Analyze each option
We will analyze the factors of each number provided in the options to see if they can be expressed in the form \(5^a 7^b\), where \(a\) and \(b\) are non-negative integers.

1. **Option 1: 35**
- Factorization: \(35 = 5^1 \times 7^1\)
- This is a factor of \(5^p 7^q\).

2. **Option 2: 175**
- Factorization: \(175 = 5^2 \times 7^1\)
- This is also a factor of \(5^p 7^q\).

3. **Option 3: 1225**
- Factorization: \(1225 = 5^2 \times 7^2\)
- This is a factor of \(5^p 7^q\).

4. **Option 4: 735**
- Factorization: \(735 = 5^1 \times 7^2 \times 3^1\)
- Here, \(735\) includes a factor of \(3\), which is not present in \(5^p 7^q\).

### Step 3: Conclusion
From the analysis, we see that \(735\) contains a factor of \(3\), which is not part of the expression \(5^p 7^q\). Therefore, **735 is not a factor of \(5^p 7^q\)**.

### Final Answer
The number that is not a factor of \(5^p 7^q\) is **735**.

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