How Many Divisors of 14! Are Perfect Cubes?

The problem of determining how many divisors of 14! are perfect cubes is an interesting exercise in number theory. Understanding this requires knowledge of prime factorization and properties of perfect cubes. Let's delve into the solution step-by-step.

Prime Factorization of 14!

The prime factorization of 14! (which is 14 factorial) is as follows:

14! 211 · 35 · 52 · 72 · 11 · 13

Identifying Perfect Cubes

A number is a perfect cube if all the exponents in its prime factorization are multiples of 3. Therefore, we need to examine the prime factors of 14! in this light.

List of Perfect Cubed Primes

29 33 23 · 33 26 26 · 33 29 · 33

These are the exponents for the primes that form perfect cubes within the factorization of 14!.

Determine the Number of Cube Divisors

To find the number of cube divisors, we "augment" each exponent (add 1) and then take the product of the augmented exponents. The augmented exponents are:

9 (from 29) 3 (from 33) 3 (from 23 · 33) 6 (from 26) 9 (from 29 · 33)

The product of the augmented exponents is:

4 · 2 8

This means there are 8 possible combinations of exponents that can form perfect cube divisors of 14!.

List the Perfect Cube Divisors

20 · 30 1 23 · 30 8 20 · 33 27 26 · 30 64 23 · 33 216 29 · 30 512 26 · 33 1728 29 · 33 13824

These are the 8 perfect cube divisors of 14!.

Clarification on the Answer

One might argue that 1 should be included as a divisor, but in the original context, it was excluded. Therefore, the correct number of divisors of 14! that are perfect cubes is 7.

Concept Highlight

The key to understanding this problem lies in recognizing the importance of prime factorization and the properties of perfect cubes. By identifying the perfect cubed components of the prime factorization, we can determine the number of perfect cube divisors through the augmented exponents method.

Related Questions

1. How many divisors of 14 are perfect cubes?

The divisors of 14 are 1, 2, 7, and 14. Among these, only 1 and 14 (2·7·7) are perfect cubes (1^3 and (2·7)^3). So there are 2 perfect cube divisors.

2. How many multiples of 14! are perfect cubes?

This is a different problem that requires a deeper understanding of the divisor function and the properties of factorial numbers.

3. What is the prime factorization of 14!

The prime factorization of 14! is 2^11 · 3^5 · 5^2 · 7^2 · 11 · 13.