Four prime numbers are written in ascending order. The product of first three numbers is 385 and last three numbers is 1001. Find the first number ?

Options

A. A)13
B. B)7
C. C)11
D. D)5

Answer

To solve the problem step by step, we will denote the four prime numbers as A, B, C, and D, arranged in ascending order.

### Step 1: Set up the equations
We know from the problem statement:
- The product of the first three prime numbers (A, B, C) is 385.
- The product of the last three prime numbers (B, C, D) is 1001.

This gives us the equations:
1. \( A \times B \times C = 385 \) (Equation 1)
2. \( B \times C \times D = 1001 \) (Equation 2)

### Step 2: Divide the two equations
To eliminate B and C, we can divide Equation 1 by Equation 2:
\[ \frac{A \times B \times C}{B \times C \times D} = \frac{385}{1001} \]
This simplifies to:
\[ \frac{A}{D} = \frac{385}{1001} \]

### Step 3: Simplify the fraction
Next, we simplify the fraction \( \frac{385}{1001} \):
- The prime factorization of 385 is \( 5 \times 7 \times 11 \).
- The prime factorization of 1001 is \( 7 \times 11 \times 13 \).

Thus, we can rewrite:
\[ \frac{385}{1001} = \frac{5 \times 7 \times 11}{7 \times 11 \times 13} = \frac{5}{13} \]

### Step 4: Establish the relationship between A and D
From the simplified fraction, we have:
\[ \frac{A}{D} = \frac{5}{13} \]
This implies:
\[ A = \frac{5}{13} \times D \]

### Step 5: Determine possible values for A and D
Since A and D are both prime numbers and must be in ascending order (A < D), we can deduce:
- If \( D = 13 \), then substituting gives \( A = 5 \).
- Since 5 is a prime number and less than 13, this is a valid solution.

### Step 6: Conclusion
Thus, the first prime number \( A \) is:
\[ \boxed{5} \]