A number theory problem by Rohith M.Athreya

Number Theory Level 3

Find the total number of positive integers which factors are of the form:

  • Exactly two factors are prime.
  • Exactly two factors are composite.
  • Exactly one factor is 1.


The answer is 0.

Rohith M.Athreya by Rohith M.Athreya , 22, India

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2 solutions

Let N be the number we are looking for.

And since N has only two prime factors let those two factors be p1 and p2

Therefore, N = ( p 1 ) m . ( p 2 ) n N=(p1)^{m} . (p2)^{n}

Hence number of factors is ( m + 1 ) ( n + 1 ) (m+1)(n+1)

But N has only 5 factors apart from itself: 1,p1,p2 and 2 composite factors.

Therefore, ( m + 1 ) ( n + 1 ) = 5 (m+1)(n+1)=5

Case 1

m + 1 = 1 m+1=1

This implies m is 0 which contradicts the fact that N has two prime factors

Similarly for case 2

n = 0 n=0 Which is also not possible therefore the number N cannot exist

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Moderator note:

I find it hard to understand what this question is asking for.

Going by Rohit's solution, it looks like you are on the right track. However, note that there is a typo when you defined N N . It should be a product, and not a sum.

Corrected it ,Sir. Thank You

Anirudh Chandramouli - 6 years ago
Rohith M.Athreya
Jun 8, 2015

the statement implies that the said number has 5 factors. 5 is prime. so,numbers with 5 factors are all power of 5 of prime numbers. however a power of five has more than two composite factors which is not unison with the question asked. thus, there exists no such number. i am not 100% sure .pls share your views if you feel the logic is flawed

0 Helpful 1 Interesting 0 Brilliant 0 Confused

What do you mean by the title of "an infinite limit"? I find it really hard to understand what you're asking for.

I have never heard of a "universal factor".

Calvin Lin Staff - 6 years ago

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1 is a universal factor of all natural numbers. an infinite limit means i am not asking for the total number of elements in a defined interval say from 1 to 100 or something of the sort instead i am asking for the total number of the numbers satisfying the defined conditions from 1 to any number n which can or may extend upto infinite bounds

Rohith M.Athreya - 6 years ago

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Thank you for the modification

Rohith M.Athreya - 6 years ago

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