Understanding the Number 2 × 3 × 5 × 7 × 11 + 1

The question asks us to determine the nature of the number formed by the product of the first five prime numbers plus one. Let's first calculate the value of this number.

The calculation is: \(2 \times 3 \times 5 \times 7 \times 11 + 1\)

Step 1: Calculate the product of the prime numbers:

• \(2 \times 3 = 6\)
• \(6 \times 5 = 30\)
• \(30 \times 7 = 210\)
• \(210 \times 11 = 2310\)

Step 2: Add 1 to the product:

\(2310 + 1 = 2311\)

So, the number in question is 2311.

Analyzing the Properties of the Number 2311

Now we need to determine if 2311 is a prime number, a power of a prime, a composite even number, or a composite odd number.

Let's define these terms:

• Prime Number: A natural number greater than 1 that has no positive divisors other than 1 and itself. Examples: 2, 3, 5, 7, 11, 13, ...
• Composite Number: A natural number greater than 1 that is not prime. It can be formed by multiplying two smaller positive integers. Examples: 4, 6, 8, 9, 10, 12, ...
• Power of a Prime: A number that can be written in the form \(p^k\), where \(p\) is a prime number and \(k\) is a positive integer (\(k \ge 1\)). Examples: \(2^3 = 8\), \(3^2 = 9\), \(5^1 = 5\), \(7^4 = 2401\).
• Even Number: An integer that is divisible by 2.
• Odd Number: An integer that is not divisible by 2.

Checking if 2311 is Even or Odd

The number 2311 ends with the digit 1. Any integer ending with 1, 3, 5, 7, or 9 is an odd number. Therefore, 2311 is an odd number.

This immediately rules out the possibility of 2311 being a composite even number (Option 3).

Checking if 2311 is a Power of a Prime

To check if 2311 is a power of a prime \(p\), we would see if \(2311 = p^k\) for some prime \(p\) and integer \(k \ge 1\).

Since 2311 is odd, it cannot be a power of 2.

Let's look at small odd prime bases:

• \(3^1=3, 3^2=9, ..., 3^7=2187, 3^8=6561\). 2311 is between \(3^7\) and \(3^8\). Not a power of 3.
• \(5^k\) must end in 5 (for \(k \ge 1\)). 2311 ends in 1. Not a power of 5.
• \(7^1=7, ..., 7^3=343, 7^4=2401\). 2311 is between \(7^3\) and \(7^4\). Not a power of 7.
• \(11^1=11, ..., 11^3=1331, 11^4=14641\). 2311 is between \(11^3\) and \(11^4\). Not a power of 11.

While a rigorous proof requires checking for other primes, 2311 does not appear to be a power of a prime number. It is very unlikely for a random number like 2311 to be a power of a prime other than the ones tested, especially considering its magnitude.

Checking if 2311 is Prime or Composite

To determine if 2311 is prime or composite, we need to check if it has any divisors other than 1 and 2311. We only need to test for prime divisors up to the square root of 2311.

The square root of 2311 is approximately \(\sqrt{2311} \approx 48.07\).

So, we need to test for divisibility by prime numbers less than or equal to 47. These primes are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.

• Divisible by 2? No (it's odd).
• Divisible by 3? Sum of digits \(2+3+1+1=7\). 7 is not divisible by 3. No.
• Divisible by 5? Does not end in 0 or 5. No.
• Divisible by 7? \(2311 = 7 \times 330 + 1\). No.
• Divisible by 11? Alternating sum of digits: \(1 - 1 + 3 - 2 = 1\). 1 is not divisible by 11. No.
• Divisible by 13? \(2311 = 13 \times 177 + 10\). No.
• Divisible by 17? \(2311 = 17 \times 135 + 16\). No.
• Divisible by 19? \(2311 = 19 \times 121 + 12\). No.
• Divisible by 23? \(2311 = 23 \times 100 + 11\). No.
• Divisible by 29? \(2311 = 29 \times 79 + 20\). No.
• Divisible by 31? \(2311 = 31 \times 74 + 17\). No.
• Divisible by 37? \(2311 = 37 \times 62 + 17\). No.
• Divisible by 41? \(2311 = 41 \times 56 + 15\). No.
• Divisible by 43? \(2311 = 43 \times 53 + 32\). No.
• Divisible by 47? \(2311 = 47 \times 49 + 8\). No.

Since 2311 is not divisible by any prime number less than or equal to its square root, it means that 2311 has no positive divisors other than 1 and 2311. Therefore, 2311 is a prime number.

Evaluating the Options

Based on our analysis that 2311 is a prime number, let's look at the options:

• Option 1: a prime number. This matches our finding.
• Option 2: not a prime, but power of a prime. This is incorrect because 2311 is prime.
• Option 3: not a power of a prime, but a composite even number. This is incorrect because 2311 is prime and odd.
• Option 4: not a power of a prime, but a composite odd number. This is incorrect because 2311 is prime.

Thus, the number 2311 is a prime number.

Revision Table: Properties of 2311

Additional Information: Primorials and Prime Numbers

The number \(2 \times 3 \times 5 \times 7 \times 11\) is an example of a primorial, denoted as \(P_k\#\). \(P_k\#\) is the product of the first \(k\) prime numbers. In this case, we calculated \(P_5\# = 2 \times 3 \times 5 \times 7 \times 11 = 2310\).

Numbers of the form \(P_k\# + 1\) are interesting in number theory. Euclid's proof for the infinitude of prime numbers considers a similar construction. If you take a finite list of primes \(\{p_1, p_2, ..., p_k\}\) and consider the number \(N = (p_1 \times p_2 \times ... \times p_k) + 1\), this number \(N\) cannot be divisible by any of the primes \(p_1, p_2, ..., p_k\). This is because dividing \(N\) by any \(p_i\) will always leave a remainder of 1.

Therefore, \(N\) must either be a new prime number itself, or it must be divisible by a prime number not in the original list. This shows there is always a prime larger than any in a finite list, proving that there are infinitely many primes.

Numbers of the form \(P_k\# + 1\) are not always prime. For instance:

• \(P_1\# + 1 = 2+1 = 3\) (prime)
• \(P_2\# + 1 = 2 \times 3 + 1 = 7\) (prime)
• \(P_3\# + 1 = 2 \times 3 \times 5 + 1 = 31\) (prime)
• \(P_4\# + 1 = 2 \times 3 \times 5 \times 7 + 1 = 211\) (prime)
• \(P_5\# + 1 = 2 \times 3 \times 5 \times 7 \times 11 + 1 = 2311\) (prime)
• \(P_6\# + 1 = 2 \times 3 \times 5 \times 7 \times 11 \times 13 + 1 = 30031 = 59 \times 509\) (composite)

The number in our question, 2311, is a prime number of this form.