Is the 11th perfect number divisible by 8?

The formula to find a perfect number is:

\[
P_n = 2^{n-1}(2^n - 1)
\]

where \(2^n - 1\) is a prime number (called a Mersenne prime). 

To find the 11th perfect number, let's first compute the corresponding Mersenne prime and perfect number for \(n = 11\).

1. \(2^{11} - 1 = 2047\)
2. We check if 2047 is prime. It is not, as \(2047 = 23 \times 89\), so \(n = 11\) does not give us a valid Mersenne prime.

To find the correct perfect number, we must use values of \(n\) corresponding to Mersenne primes. For the 11th perfect number, the Mersenne prime is associated with \(n = 61\) because \(2^{61} - 1\) is a prime number.

The 11th perfect number is:

\[
P_{11} = 2^{60}(2^{61} - 1)
\]

Now, to check if the 11th perfect number is divisible by 8:

\[
P_{11} = 2^{60} \times (2^{61} - 1)
\]

Since \(P_{11}\) contains the factor \(2^{60}\), it is divisible by \(2^3 = 8\).

Thus, **the 11th perfect number is divisible by 8**.

---

Would you like any additional details or have any further questions?

Here are 5 related questions:

1. What is the smallest perfect number, and why is it perfect?
2. How are Mersenne primes related to perfect numbers?
3. What are the properties of an even perfect number?
4. Is there an odd perfect number, or has one been found?
5. How can you prove that a given number is a perfect number?

**Tip:** When a number is a power of 2, it will always be divisible by smaller powers of 2, including 8.

Explore whether the 11th perfect number is divisible by 8. This problem involves number theory concepts like perfect numbers and Mersenne primes, using the formula P_n = 2^{n-1}(2^n - 1). Learn how divisibility by powers of 2 applies to large perfect numbers. Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation