December 14th, 2016, 11:19 PM  #1 
Senior Member Joined: Apr 2008 Posts: 176 Thanks: 3  Is there a way to find the positive integers with 4 factors less than 100?
I know how to do the problem below in a tedious way. But I cannot think of an easier way to do it. (ex) Consider all the positive integers less than 100 that have exactly four factors which include 1 and the number itself. Find the sum of all these integers. The most obvious way is to list them all out and add them up. 6 8 10 14 15 21 22 26 27 33 ... Could someone show me an easier way to do it? Thank you very much. 
December 15th, 2016, 12:00 AM  #2 
Senior Member Joined: Sep 2015 From: Southern California, USA Posts: 1,495 Thanks: 753 
can the factors be composite numbers?

December 15th, 2016, 06:32 AM  #3 
Global Moderator Joined: Dec 2006 Posts: 18,052 Thanks: 1395 
The examples include 27, where 9 is a composite factor.

December 15th, 2016, 07:08 AM  #4 
Senior Member Joined: Sep 2015 From: Southern California, USA Posts: 1,495 Thanks: 753  
December 15th, 2016, 11:18 AM  #5 
Banned Camp Joined: Jun 2014 From: Earth Posts: 945 Thanks: 191  
December 15th, 2016, 11:53 AM  #6 
Senior Member Joined: Sep 2015 From: Southern California, USA Posts: 1,495 Thanks: 753  
December 15th, 2016, 12:03 PM  #7  
Senior Member Joined: Sep 2015 From: Southern California, USA Posts: 1,495 Thanks: 753  Quote:
as every number has 1 and itself as factors, and as any number of factors can be combined to result in 2 factors, you are looking for all the numbers between 4 and 100 that are not primes. The sum of all integers 1 to 100 is $50\cdot 101 = 5050$ Subtract off the the sum of the first 3 integers, i.e. $6$ to get $5044$ Now subtract off the sum of the primes $p,~4 < p < 100$ I don't know any fancy way of doing this but there's only 25 of them. They sum to $1055$ so we end up with $50441055 = 3989$  
December 16th, 2016, 02:48 PM  #8 
Senior Member Joined: Apr 2008 Posts: 176 Thanks: 3 
Hi everyone. I am sorry. I tried to do the problem mentally as I was looking for those positive integers with only four factors. I accidentally picked "27" without realizing that it was composite. Composite numbers are not allowed. I am so sorry. Thanks for your solution, romsek. How do you remove all the composite numbers from the list? Thanks again. Last edited by davedave; December 16th, 2016 at 02:53 PM. 
December 16th, 2016, 04:06 PM  #9  
Senior Member Joined: Sep 2015 From: Southern California, USA Posts: 1,495 Thanks: 753  Quote:
Well now what you are looking at is all unordered pairs of primes whose product is 100 or less. So 2x2 through 2x47, 3x3 through 3x31, 5x5 through 5x19, etc up to 7x13 Is there a clever way to sum these? Dunno. I'm not seeing one jump out at me.  
December 16th, 2016, 04:26 PM  #10 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,353 Thanks: 464 Math Focus: Yet to find out. 
Poor romsek xD


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100, factors, find, integers, positive, positives 
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