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December 15th, 2016, 12:19 AM   #1
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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.
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December 15th, 2016, 01:00 AM   #2
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can the factors be composite numbers?
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December 15th, 2016, 07:32 AM   #3
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The examples include 27, where 9 is a composite factor.
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December 15th, 2016, 08:08 AM   #4
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Quote:
Originally Posted by skipjack View Post
The examples include 27, where 9 is a composite factor.
yeah I saw that which is why I asked, I doubt it should be there.
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December 15th, 2016, 12:18 PM   #5
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Quote:
Originally Posted by skipjack View Post
The examples include 27, where 9 is a composite factor.
The examples also include 8, where 4 is a composite factor.
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December 15th, 2016, 12:53 PM   #6
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The examples also include 8, where 4 is a composite factor.
well alrighty then
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December 15th, 2016, 01:03 PM   #7
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Quote:
Originally Posted by davedave View Post
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.
since composite numbers are allowed...

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 $5044-1055 = 3989$
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December 16th, 2016, 03:48 PM   #8
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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 03:53 PM.
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December 16th, 2016, 05:06 PM   #9
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Quote:
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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.
grumble I figured as much.

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.
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December 16th, 2016, 05:26 PM   #10
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Poor romsek xD
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