Answer:
Step-by-step explanation:
Hello,
What other numbers are of this form have exactly four factors?
[tex]8=2^3\text{ has 4 factors } 1, 2, 2^2, 2^3\\\\\text{For any prime p, }p^3\text{ has 4 factors } 1, p, p^2, p^3\\\\[/tex]
Do all the perfect cubes have exactly four factors?
[tex]\text{If p is not prime, then it is not true anymore} \\ \\\text{For instance, }8^3\text{ has more than 4 factors } 1, 2^2, 2^3, 2^4, etc..[/tex]
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.
We know that any number can be decomposed as a product of prime numbers.
For example:
[tex]8 = 1\cdot 2\cdot 2\cdot 2[/tex]
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:
[tex]X = n\cdot n\cdot n \cdot 1[/tex]
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:
[tex]8\cdot 8\cdot8= 512[/tex]
512 is a perfect cube, and if we decompose it we get:
[tex]512 = 8\cdot 8\cdot 8 = (2\cdot 2\cdot 2)\cdot (2\cdot 2\cdot 2)\cdot (2\cdot 2\cdot 2)\cdot 1[/tex]
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.
[tex]1\cdot 2\cdot 3\cdot 5 = 30[/tex]
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:
https://brainly.com/question/832797
