Identifying Consecutive Prime Numbers

Let the four consecutive prime numbers be denoted by $p_1$, $p_2$, $p_3$, and $p_4$, where $p_1 < p_2 < p_3 < p_4$.

Using Product Information

We are given two pieces of information:

• The product of the last three primes is 7429: $p_2 \times p_3 \times p_4 = 7429$
• The product of the first three primes is 4199: $p_1 \times p_2 \times p_3 = 4199$

Calculating Prime Number Ratio

To find the relationship between the first and the last prime number, we can divide the two given products:

$ \frac{p_2 \times p_3 \times p_4}{p_1 \times p_2 \times p_3} = \frac{7429}{4199} $

By canceling the common terms ($p_2$ and $p_3$), we get:

$ \frac{p_4}{p_1} = \frac{7429}{4199} $

Prime Factorization Method

Alternatively, we can find the prime factors of the given products.

Factorizing 4199:

$ 4199 = 13 \times 323 = 13 \times 17 \times 19 $

So, the first three consecutive primes ($p_1, p_2, p_3$) are 13, 17, and 19.

Factorizing 7429:

Since $p_2=17$ and $p_3=19$, we can find $p_4$ using the second product:

$ 17 \times 19 \times p_4 = 7429 $

$ 323 \times p_4 = 7429 $

$ p_4 = \frac{7429}{323} $

$ p_4 = 23 $

The number 23 is a prime number.

Determining the Largest Prime

The four consecutive prime numbers are 13, 17, 19, and 23.

The largest of these prime numbers is 23.