Question

What are the numbers that have exactly 5 factors?

Answer 1

`120`

Explanation:

Because when you look for the minimum number with five factors you have perform `5!` (`5` factorial). This is when all number from 1 to 5 are multiplied together, which means that all of the five numbers are factors of the result number.

`1*2*3*4*5=5! =120`

Answer 2

Numbers that can be expressed as `x^4` where `x>1` and `x` is prime.

Explanation:

Numbers that have a chance of having exactly 5 factors have to be perfect squares (since factors come in pairs, such as with 12: 1, 12; 2, 6; 3, 4) so that they are multiplied by one of their factors twice.

https://socratic.org/questions/eduardo-thinks-of-a-number-between-1-and-20-that-has-exactly-5-factors-what-numb#343583

16 is the first number where this happens and results in 5 factors: 1, 16; 2, 8; 4

We can check other perfect squares:

`25 => 1, 25; 5 color(white)(000)color(red)X`

`36 => 1, 36; 2, 18; 3, 12;... color(white)(000)color(red)X`

`49 => 1, 49; 7 color(white)(000)color(red)X`

`64 => 1, 64; 2, 32; 4,16; ... color(white)(000)color(red)X`

`81 => 1, 81; 3,27; 9 color(white)(000)color(green)root`

And I think we've found the pattern - the only numbers with exactly 5 factors are those that can be expressed as `x^4, x>1`, and `x` is prime:

`16=2^4, 81=3^4`

Put another way, the factors will follow this pattern:

`x^0 xx x^4, x^1 xx x^3, (x^2)^2`

For 16, we have `2^0 xx 2^4=1xx16; 2^1 xx 2^3=2xx8; (x^2)^2=(2^2)^2=4^2=16`

And so I'd expect the next few numbers to be `5^4=625, 7^4=2401, and 11^4=14,641`