Which number should replace both the asterisks in (*/21) `xx` (*/(189)) = 1 ?

Options

A. 21
B. 63
C. 3969
D. 147

Answer

To solve the equation \( \frac{*}{21} \times \frac{*}{189} = 1 \), we need to find the number that can replace both asterisks.

### Step-by-step Solution:

1. **Set up the equation**:
We start with the equation:
\[ \frac{x}{21} \times \frac{x}{189} = 1 \]

2. **Multiply the fractions**:
When we multiply the two fractions, we get:
\[ \frac{x \times x}{21 \times 189} = 1 \]
This simplifies to:
\[ \frac{x^2}{21 \times 189} = 1 \]

3. **Cross-multiply**:
To eliminate the fraction, we cross-multiply:
\[ x^2 = 21 \times 189 \]

4. **Calculate \( 21 \times 189 \)**:
Now we need to calculate the product:
\[ 21 \times 189 = 3969 \]

5. **Take the square root**:
To find \( x \), we take the square root of both sides:
\[ x = \sqrt{3969} \]

6. **Calculate the square root**:
We can simplify \( \sqrt{3969} \).
By prime factorization, we find:
\[ 3969 = 3^4 \times 7^2 \]
Therefore:
\[ \sqrt{3969} = 3^2 \times 7 = 9 \times 7 = 63 \]

7. **Final answer**:
Thus, the number that should replace both asterisks is:
\[ x = 63 \]

### Summary:
The number that replaces both asterisks in the equation \( \frac{*}{21} \times \frac{*}{189} = 1 \) is **63**.