If m and n are positive integers and `(2^(18))(5^m) = (20^n)`, what is the value of m?

Answer

To solve the equation \( (2^{18})(5^m) = (20^n) \), we will start by expressing \( 20 \) in terms of its prime factors.

### Step 1: Express \( 20 \) in terms of its prime factors
We know that:
\[ 20 = 2^2 \times 5^1 \]
Thus, we can rewrite \( 20^n \) as:
\[ 20^n = (2^2 \times 5^1)^n = 2^{2n} \times 5^n \]

### Step 2: Set the equation
Now we can rewrite the original equation:
\[ (2^{18})(5^m) = 2^{2n} \times 5^n \]

### Step 3: Equate the powers of the same bases
Since the bases are the same, we can equate the powers of \( 2 \) and \( 5 \) separately.

For the base \( 2 \):
\[ 18 = 2n \quad \text{(1)} \]

For the base \( 5 \):
\[ m = n \quad \text{(2)} \]

### Step 4: Solve for \( n \) from equation (1)
From equation (1):
\[ n = \frac{18}{2} = 9 \]

### Step 5: Substitute \( n \) back into equation (2)
Now substituting \( n = 9 \) into equation (2):
\[ m = n = 9 \]

### Conclusion
Thus, the value of \( m \) is:
\[ \boxed{9} \]