Question

Eduardo thinks of a number between 1 and 20 that has exactly 5 factors. What number is he thinking of?

Answer 1

Answer is `16`

Explanation:

It is apparent that a number cannot be a prime number, if it has exactly `5` factors. So it is among `{4,6,8,10,12,14,16,18,20}`

As a product of two prime numbers say `p_1` and `p_2`, will have just four factors `{1,p_1,p_2,p_1xxp_2}`, both for `p_1=p_2` (for this we will have just three factors) and `p_1!=p_2`, we also rule out `{4,6,10,14}`.

Now for remaining `{8,12,16,18,20}`

`8` has four factors `{1,2,4,8}`; `12` has six factors `{1,2,3,4,6,12}`; `16` has five factors `{1,2,4,8,16}`: `18` has six factors `{1,2,3,6,9,18}` and `20` has six factors `{1,2,4,5,10,20}`.

Hence answer is `16`

Answer 2

`16` has `5` factors

Explanation:

The most direct method of finding the number that has `5` factors depends on knowing that square numbers have an ODD number of factors.

This is because factors are always in pairs, but in a square, one of the factors is multiplied by itself and it is only counted once. (the square root)

This means that of all the numbers from `1 " to "20`, the only numbers we need to look at again are the square numbers:

`1," "4," "9," "16`

It would seem that the largest one is most likely to have `5` factors, so let's look at `16` first.

Factors of `16:" "1," " 2," " 4," " 8," " 16" "larr` there are `5` factors!

As a check, let's consider the other squares:

`1` has only `1` factor.
`4` has `3` factors: `1," "2," "4`
`9` has `3` factors: `1," "3," "9`