Four prime numbers are written in ascending order of their magnitudes. The product of the first three is 715 and that of the last three is 2431. What is the largest given prime number?

Options

A. 5
B. 19
C. 17
D. 23

Answer

To solve the problem, we need to find four prime numbers \(a\), \(b\), \(c\), and \(d\) such that:

1. The product of the first three primes \(a \times b \times c = 715\)
2. The product of the last three primes \(b \times c \times d = 2431\)

We will follow these steps:

### Step 1: Factor the products
First, we need to factor the two products to find the prime numbers.

**For 715:**
- Start dividing by the smallest prime numbers:
- \(715 \div 5 = 143\) (since 715 ends with a 5, it's divisible by 5)
- Now factor 143:
- \(143 \div 11 = 13\) (143 is divisible by 11)

So, the prime factorization of 715 is:
\[ 715 = 5 \times 11 \times 13 \]

**For 2431:**
- Start dividing by the smallest prime numbers:
- \(2431 \div 11 = 221\) (since 2431 is divisible by 11)
- Now factor 221:
- \(221 \div 13 = 17\) (221 is divisible by 13)

So, the prime factorization of 2431 is:
\[ 2431 = 11 \times 13 \times 17 \]

### Step 2: Identify the common primes
From the factorizations:
- \(a \times b \times c = 5 \times 11 \times 13\)
- \(b \times c \times d = 11 \times 13 \times 17\)

We can see that \(b\) and \(c\) are common in both products. Thus, we can denote:
- \(b = 11\)
- \(c = 13\)

### Step 3: Find \(a\) and \(d\)
Now, substituting \(b\) and \(c\) back into the equations:

1. From \(a \times b \times c = 715\):
\[ a \times 11 \times 13 = 715 \]
\[ a \times 143 = 715 \]
\[ a = \frac{715}{143} = 5 \]

2. From \(b \times c \times d = 2431\):
\[ 11 \times 13 \times d = 2431 \]
\[ 143 \times d = 2431 \]
\[ d = \frac{2431}{143} = 17 \]

### Step 4: List the prime numbers
Now we have:
- \(a = 5\)
- \(b = 11\)
- \(c = 13\)
- \(d = 17\)

### Step 5: Identify the largest prime number
The largest prime number among \(5\), \(11\), \(13\), and \(17\) is:
\[ \text{Largest prime number} = 17 \]

### Final Answer
The largest given prime number is \( \boxed{17} \).