On multiplying a number by 7, all the digits in the product appear as 3's. the smallest such number is

Options

A. 47619
B. 47719
C. 47649
D. 48619

Answer

To solve the problem of finding the smallest number that, when multiplied by 7, results in a product consisting solely of the digit '3', we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**: We need to find a number \( x \) such that \( 7 \times x \) results in a number made entirely of the digit '3'. This means the product can be represented as \( 333...3 \) (a series of threes).

2. **Formulating the Product**: The product can be expressed in a mathematical form. A number made up entirely of threes can be represented as:
\[ 3 \times (10^n - 1) / 9 \]
where \( n \) is the number of threes. This is derived from the fact that \( 333...3 \) can be rewritten as \( 3 + 30 + 300 + ... \).

3. **Setting Up the Equation**: We want:
\[ 7x = 3 \times \frac{10^n - 1}{9} \]
Rearranging gives:
\[ x = \frac{3 \times (10^n - 1)}{63} \]
Simplifying:
\[ x = \frac{10^n - 1}{21} \]

4. **Finding the Smallest \( n \)**: We need \( 10^n - 1 \) to be divisible by 21. We can check for the smallest \( n \) by calculating \( 10^n \mod 21 \).

5. **Calculating \( 10^n \mod 21 \)**:
- \( n = 1: 10^1 \mod 21 = 10 \)
- \( n = 2: 10^2 \mod 21 = 16 \)
- \( n = 3: 10^3 \mod 21 = 13 \)
- \( n = 4: 10^4 \mod 21 = 4 \)
- \( n = 5: 10^5 \mod 21 = 19 \)
- \( n = 6: 10^6 \mod 21 = 1 \)

We find that \( 10^6 \equiv 1 \mod 21 \). This means \( 10^6 - 1 \) is divisible by 21.

6. **Calculating \( x \)**: Now substituting \( n = 6 \):
\[ x = \frac{10^6 - 1}{21} = \frac{1000000 - 1}{21} = \frac{999999}{21} = 47619 \]

7. **Verification**: Finally, we verify:
\[ 7 \times 47619 = 333333 \]
which consists entirely of the digit '3'.

### Conclusion:
The smallest number \( x \) such that \( 7 \times x \) results in a number made entirely of the digit '3' is:
\[ \boxed{47619} \]