Slightly Prime?

Number Theory Level 4

A number N N is called Slightly Prime if it has exactly 3 3 factors.

How many 3 3 digit Slightly Prime numbers exist?


The answer is 7.

Swapnil Das by Swapnil Das , 20, India

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2 solutions

Sam Bealing
Apr 15, 2016

For a number to have an odd number of factors it must be a perfect square (otherwise all the factors have a pair and so it would have an even number of factors).

As we are looking for only three factors, the number must be a square of a prime p p and hence have divisors 1 , p , p 2 1,p,p^2 .

In order for 100 ≤ p 2 ≤ 999 100 \leq p^2 \leq 999 , p p must be between 10 ≤ p ≤ 31 10 \leq p \leq 31 giving p = 11 , 13 , 17 , 19 , 23 , 29 , 31 p=11,13,17,19,23,29,31 hence giving the numbers as 121 , 169 , 289 , 361 , 529 , 841 , 961 121,169, 289, 361, 529, 841, 961

This gives an answer of 7 \boxed{7}

4 Helpful 0 Interesting 0 Brilliant 0 Confused

Moderator note:

Great explanation combining the facts about number of divisors and restricting the range of primes.

Excellent solution. Pardon me for the inconvenience caused.

Swapnil Das - 5 years, 2 months ago

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I'd seen this question before. 105 points for free! This is more of a counting question though

Mehul Arora - 5 years, 2 months ago

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True. You know, though I made this question myself, I found the similar version question in the FIITJEE material. And the answer given was 21 ---__---

Swapnil Das - 5 years, 2 months ago

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@Swapnil Das – Hahahaha, Happens. It could be a different question though. Also, this question existed wayyyyy back.

Mehul Arora - 5 years, 2 months ago

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@Mehul Arora – Nah, it was exactly the same. You can check level 3 objective problems.

Swapnil Das - 5 years, 2 months ago

Did the same

Aditya Kumar - 5 years, 1 month ago
Tran Hieu
Apr 15, 2016

If he change the definition to "exactly 3 factors" not counting itself then the problem is very interesting

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Then the number would have to be a product of two primes and it would just become a case of counting the possible products of two primes that result in a three-digit number.

And also you would have to count three digit cubes of prime numbers.

Sam Bealing - 5 years, 2 months ago

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