If x, y and z are integers, is x even?
(1) `10^(x)=(4^(y))(5^(z))`
(2) `3^(x+5)=27^(y+1)`

Answer

To determine if \( x \) is even given the statements, we will analyze each statement step by step.

### Step 1: Analyze Statement (1)
We start with the equation from statement (1):
\[ 10^x = 4^y \cdot 5^z \]
We can express \( 10 \) and \( 4 \) in terms of their prime factors:
\[ 10 = 2 \cdot 5 \quad \text{and} \quad 4 = 2^2 \]
Thus, we can rewrite the equation as:
\[ (2 \cdot 5)^x = (2^2)^y \cdot 5^z \]
This expands to:
\[ 2^x \cdot 5^x = 2^{2y} \cdot 5^z \]
Now, we can equate the powers of the same bases:
1. For base \( 2 \):
\[ x = 2y \quad \text{(1)} \]
2. For base \( 5 \):
\[ x = z \quad \text{(2)} \]

From equation (1), since \( x = 2y \), we can see that \( x \) is a multiple of \( 2 \). Therefore, \( x \) is guaranteed to be even regardless of whether \( y \) is even or odd.

### Step 2: Analyze Statement (2)
Now, we analyze statement (2):
\[ 3^{x+5} = 27^{y+1} \]
We can express \( 27 \) as \( 3^3 \):
\[ 3^{x+5} = (3^3)^{y+1} \]
This simplifies to:
\[ 3^{x+5} = 3^{3(y+1)} \]
Now, since the bases are the same, we can equate the exponents:
\[ x + 5 = 3(y + 1) \]
Expanding this gives:
\[ x + 5 = 3y + 3 \]
Rearranging leads to:
\[ x = 3y - 2 \quad \text{(3)} \]

From equation (3), we cannot definitively conclude whether \( x \) is even or odd. For example:
- If \( y = 1 \), then \( x = 3(1) - 2 = 1 \) (odd).
- If \( y = 2 \), then \( x = 3(2) - 2 = 4 \) (even).

Thus, statement (2) does not provide a clear answer regarding whether \( x \) is even.

### Conclusion
From our analysis:
- **Statement (1)** guarantees that \( x \) is even.
- **Statement (2)** does not provide sufficient information to determine if \( x \) is even.

Therefore, the answer to the question "Is \( x \) even?" is **Yes**, based on statement (1).