Question

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?

Answer 1

Answer:

Step-by-step explanation:

Hello,

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

$8=2^3\text{ has 4 factors } 1, 2, 2^2, 2^3\\\\\text{For any prime p, }p^3\text{ has 4 factors } 1, p, p^2, p^3$

Do all the perfect cubes have exactly four factors?

$\text{If p is not prime, then it is not true anymore} \\ \\\text{For instance, }8^3\text{ has more than 4 factors } 1, 2^2, 2^3, 2^4, etc..$

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.