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

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69
Nov 29, 2016

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 $\left\{4 , 6 , 8 , 10 , 12 , 14 , 16 , 18 , 20\right\}$

As a product of two prime numbers say ${p}_{1}$ and ${p}_{2}$, will have just four factors $\left\{1 , {p}_{1} , {p}_{2} , {p}_{1} \times {p}_{2}\right\}$, both for ${p}_{1} = {p}_{2}$ (for this we will have just three factors) and ${p}_{1} \ne {p}_{2}$, we also rule out $\left\{4 , 6 , 10 , 14\right\}$.

Now for remaining $\left\{8 , 12 , 16 , 18 , 20\right\}$

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

Hence answer is $16$

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20
May 18, 2017

$16$ 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 $1 \text{ to } 20$, the only numbers we need to look at again are the square numbers:

$1 , \text{ "4," "9," } 16$

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

Factors of $16 : \text{ "1," " 2," " 4," " 8," " 16" } \leftarrow$ there are $5$ factors!

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

$1$ has only $1$ factor.
$4$ has $3$ factors: $1 , \text{ "2," } 4$
$9$ has $3$ factors: $1 , \text{ "3," } 9$

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