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A positive integer is called nice if it is a multiple of $6.$

 

A certain nice positive integer $n$ has exactly $8$ positive divisors.  How many prime numbers are divisors of $n?$

 Jun 23, 2024

Best Answer 

 #1
avatar+1230 
+1

If a number is divisble by 6, then it is also divisble by \(1,2,3,6\)

 

Now, let's take a look at factors of 6. 

We know that \(12\) has 6 factors, as we have \(1,2,3,4,6,12\)

18 also 6 factors, as we have \(1,2,3,6,9,18\)

 

However, 24 has exactly 8 factors. We have

\(1,2,3,4,6,8,12,24\)

 

The prime factors are \(2,3\), so our answer is just 2. 

 

So there are 2 prime factors. 

 

Thanks! :)

 Jun 23, 2024
 #1
avatar+1230 
+1
Best Answer

If a number is divisble by 6, then it is also divisble by \(1,2,3,6\)

 

Now, let's take a look at factors of 6. 

We know that \(12\) has 6 factors, as we have \(1,2,3,4,6,12\)

18 also 6 factors, as we have \(1,2,3,6,9,18\)

 

However, 24 has exactly 8 factors. We have

\(1,2,3,4,6,8,12,24\)

 

The prime factors are \(2,3\), so our answer is just 2. 

 

So there are 2 prime factors. 

 

Thanks! :)

NotThatSmart Jun 23, 2024

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