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 and ensure it can be expressed in the form \( 2^m \times 5^n \), where \( m \) and \( n \) are non-negative integers.

### Step-by-Step Solution:

1. **Identify the condition for a terminating decimal**:
A fraction is a terminating decimal if its denominator (in simplest form) can be expressed as \( 2^m \times 5^n \).

2. **Factor the denominator**:
We start with the denominator \( 3500 \). We can factor it as follows:
\[ 3500 = 35 \times 100 = 35 \times (10^2) = 35 \times (2^2 \times 5^2) \]
Now, factor \( 35 \):
\[ 35 = 7 \times 5 \]
Therefore, we can express \( 3500 \) as:
\[ 3500 = 7 \times 5^3 \times 2^2 \]

3. **Factor the numerator**:
Next, we factor the numerator \( 891 \):
\[ 891 = 9 \times 99 = 9 \times (9 \times 11) = 9^2 \times 11 = 3^4 \times 11 \]

4. **Combine the fraction**:
Now we can write the fraction:
\[ \frac{891}{3500} = \frac{3^4 \times 11}{7 \times 5^3 \times 2^2} \]

5. **Identify the prime factors**:
The prime factors in the denominator are \( 2^2 \), \( 5^3 \), and \( 7 \). For the fraction to be a terminating decimal, we need to eliminate the factor \( 7 \) from the denominator.

6. **Multiply by the necessary factor**:
To eliminate \( 7 \) from the denominator, we must multiply the fraction by \( 7 \):
\[ \frac{891 \times 7}{3500 \times 7} = \frac{3^4 \times 11 \times 7}{5^3 \times 2^2} \]
Now, the denominator is \( 5^3 \times 2^2 \), which is in the required form \( 2^m \times 5^n \).

7. **Conclusion**:
The smallest number by which \( \frac{891}{3500} \) must be multiplied to make it a terminating decimal is \( 7 \).

### Final Answer:
The smallest number is \( 7 \).