What is the smallest number by which `(891)/(3500)` must be multiplied so it becomes a terminating decimal?

Options

A. 6
B. 7
C. 10
D. 5

Answer

To determine the smallest number by which \(\frac{891}{3500}\) must be multiplied so that it becomes a terminating decimal, we need to analyze the denominator \(3500\) and its prime factorization.

### Step 1: Factor the denominator
First, we factor \(3500\):
\[ 3500 = 35 \times 100 \]
Next, we factor \(35\) and \(100\):
\[ 35 = 7 \times 5 \quad \text{and} \quad 100 = 10^2 = (2 \times 5)^2 = 2^2 \times 5^2 \]
Combining these, we get:
\[ 3500 = 7 \times 5 \times 2^2 \times 5^2 = 7 \times 2^2 \times 5^3 \]

### Step 2: Identify the conditions for a terminating decimal
A fraction \(\frac{A}{B}\) is a terminating decimal if the denominator \(B\) (after simplification) has only the prime factors \(2\) and \(5\). In our case, the denominator \(3500\) has an extra factor of \(7\).

### Step 3: Determine the necessary multiplication
To eliminate the factor of \(7\) from the denominator, we need to multiply the fraction by \(7\) so that \(7\) cancels out:
\[ \frac{891 \times 7}{3500 \times 7} = \frac{891 \times 7}{7 \times 2^2 \times 5^3} = \frac{891 \times 7}{2^2 \times 5^3} \]
Now the denominator is only composed of the factors \(2\) and \(5\), which means it can be expressed as a terminating decimal.

### Step 4: Conclusion
Thus, the smallest number by which \(\frac{891}{3500}\) must be multiplied to become a terminating decimal is:
\[ \boxed{7} \]