Alana noticed that a number like 8 = 2 (i. e. ) has exactly four factors: 1, 2, 2 , and ! 2 . What other numbers are of this form have exactly four factors? Do all the perfect cubes have exactly four factors? Is it true that all numbers that have exactly four factors are perfect cubes?

We know that any number can be decomposed as a product of prime numbers.

For example:

8 has four factors if we count the 1.

Now we need to answer some questions regarding this:

What other numbers are of this form have exactly four factors?

8 is a cube number.

Remember that all cube numbers X can be written as:

Now, if n is prime, then we can say that X is of the form of 8 and also has 4 factors.

But let's see the case:

512 is a perfect cube, and if we decompose it we get:

So we found a perfect cube that has 10 factors.

This means that not all perfect cubes have exactly 4 factors.

Now, all numbers that have exactly four factors are perfect cubes?

This is also false, just let's create a counterexample.

the number 30 is not a perfect cube, and we can see that it has exactly 4 factors.

If you want to learn more, you can read:

link

Step-by-step explanation:

Hello,

What other numbers are of this form have exactly four factors?

Do all the perfect cubes have exactly four factors?

Is it true that all numbers that have exactly four factors are perfect cubes?

6 has 4 factors : 1, 2, 3, 6 but this is not a perfect cube.

Thank you.