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# Number Theory

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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

#1
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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
+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! :)

NotThatSmart Jun 23, 2024