A positive integer P ,when multiplied by 9 gives a result which consists of 5 followed by some number of 7 's. What is the number of 7 's in the product thus obtained , if P is the least such number possible ?

Options

A. 16
B. 11
C. 9
D. 7

Answer

To solve the problem step by step, we need to find the smallest positive integer \( P \) such that when multiplied by 9, the result is a number that starts with 5 followed by some number of 7s.

### Step 1: Understand the Form of the Product
The product \( 9P \) must be of the form \( 5\underbrace{77\ldots7}_{n \text{ times}} \), where \( n \) is the number of 7s following the digit 5.

### Step 2: Calculate the Sum of Digits
For \( 9P \) to be divisible by 9, the sum of its digits must also be divisible by 9. The sum of the digits of the number \( 5\underbrace{77\ldots7}_{n \text{ times}} \) can be expressed as:
\[ \text{Sum of digits} = 5 + 7n \]
We need to find the smallest \( n \) such that \( 5 + 7n \) is divisible by 9.

### Step 3: Check Values of \( n \)
We will check values of \( n \) starting from 1:

- **For \( n = 1 \)**:
\[ 5 + 7 \times 1 = 12 \quad (\text{not divisible by } 9) \]

- **For \( n = 2 \)**:
\[ 5 + 7 \times 2 = 19 \quad (\text{not divisible by } 9) \]

- **For \( n = 3 \)**:
\[ 5 + 7 \times 3 = 26 \quad (\text{not divisible by } 9) \]

- **For \( n = 4 \)**:
\[ 5 + 7 \times 4 = 33 \quad (\text{not divisible by } 9) \]

- **For \( n = 5 \)**:
\[ 5 + 7 \times 5 = 40 \quad (\text{not divisible by } 9) \]

- **For \( n = 6 \)**:
\[ 5 + 7 \times 6 = 47 \quad (\text{not divisible by } 9) \]

- **For \( n = 7 \)**:
\[ 5 + 7 \times 7 = 54 \quad (\text{divisible by } 9) \]

### Step 4: Conclusion
The smallest \( n \) for which \( 5 + 7n \) is divisible by 9 is \( n = 7 \). Therefore, the product \( 9P \) is \( 57777777 \) (5 followed by 7 sevens).

Thus, the number of 7s in the product obtained is **7**.

### Final Answer
The number of 7s in the product is **7**.