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If x is the smallest positive integer that is not prime

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If x is the smallest positive integer that is not prime

by melguy » Sat Aug 24, 2013 5:06 am
I am confused with the language of the problem. What is the key takeaway from this problem and how can I apply it on similar problems e.g. if we used 12! or 20! instead of 50!

In the case of 12! will the value be 13 x 2 = 26
In the case of 20! will the value be 23 x 2 = 46

Please help me with the problem. Thanks
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by Brent@GMATPrepNow » Sat Aug 24, 2013 6:33 am
melguy wrote:I am confused with the language of the problem. What is the key takeaway from this problem and how can I apply it on similar problems e.g. if we used 12! or 20! instead of 50!

In the case of 12! will the value be 13 x 2 = 26
In the case of 20! will the value be 23 x 2 = 46

Please help me with the problem. Thanks
Looks good, melguy.

EDIT: My original post had an error in it that azpire points out later in this thread. So, I've had to add a proviso to the general rule:

If x is the smallest positive non-prime integer that is not a factor of k!, then x = (the smallest prime that's greater than k)(2) AS LONG AS k > 6
If k = 1, then 4 is smallest positive non-prime integer that is not a factor of k!
If k = 2, then 4 is smallest positive non-prime integer that is not a factor of k!
If k = 3, then 4 is smallest positive non-prime integer that is not a factor of k!
If k = 4, then 9 is smallest positive non-prime integer that is not a factor of k!
If k = 5, then 9 is smallest positive non-prime integer that is not a factor of k!


Some examples
- The smallest positive non-prime integer that is not a factor of 14! is (17)(2)
- The smallest positive non-prime integer that is not a factor of 60! is (61)(2)
- The smallest positive non-prime integer that is not a factor of 50! is (53)(2).

Cheers,
Brent
Last edited by Brent@GMATPrepNow on Sun Apr 20, 2014 7:45 am, edited 1 time in total.
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by ganeshrkamath » Sat Aug 24, 2013 7:54 am
melguy wrote:I am confused with the language of the problem. What is the key takeaway from this problem and how can I apply it on similar problems e.g. if we used 12! or 20! instead of 50!

In the case of 12! will the value be 13 x 2 = 26
In the case of 20! will the value be 23 x 2 = 46

Please help me with the problem. Thanks
Edit: Got it wrong the first time.

I agree with Brent's solution.

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by Brent@GMATPrepNow » Sat Aug 24, 2013 8:03 am
ganeshrkamath wrote: Every positive integer less than or equal to 50 is a factor of 50!
The next composite number is 51.

Factors of 51: 1,3,17,51
Sum of the factors of 51 = 1+3+17+51 = 72

Choose C

Cheers
Be careful - 51 is a factor of 50!
51 = (3)(17)
50! = (50)(49)(48)(47)....(17)(16)....(4)(3)(2)(1)

On the other hand, we know that the prime number 53 cannot be a factor of 50!, since there is no way to "create" 53 with any of the numbers from the product 50!
So, (53)(2) will be the smallest non-prime integer that is not a factor of 50!

(53)(2) = 106
The factors of 106 are 1, 2, 53 and 106
Their sum = 1 + 2 + 53 + 106 = 162 = D

Cheers,
Brent
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by vinay1983 » Sat Aug 24, 2013 9:31 am
I did not get anyone of you :(
You can, for example never foretell what any one man will do, but you can say with precision what an average number will be up to!
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by ganeshrkamath » Sat Aug 24, 2013 9:39 am
vinay1983 wrote:I did not get anyone of you :(
The problem asks us to find the least composite that is not a factor of 50!
Now every number from 1 to 50 is a factor of 50!
In addition, every product combination of any of the numbers from 1 to 50 is a factor of 50!
(for example, 51 = 3*17 => since 50! = 50*49*48*...*17*...*3*2*1, 3*17 is a factor of 50!)

Now for a composite number to not be a factor of 50!, it should consist of a prime number >50.
The smallest such composite number is 53*2 (the 2 makes it a composite number).

Hope this helps.
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by Brent@GMATPrepNow » Sat Aug 24, 2013 9:54 am
vinay1983 wrote:I did not get anyone of you :(
Here's the original question:

If x is the smallest positive integer that is not prime and not a factor of 50!, what is the sum of the factors of x?
A) 51
B) 54
C) 72
D) 162
E) 50! + 2

Let's first find the smallest number (prime or not prime) that is not a factor of 50!
Since 50! = (50)(49)(48)(47).....(4)(3)(2)(1), we can see that 50, 49, 48 (etc) are all factors of 50!

What about 51?
Notice that 50! = (50)(49)(48)(47)....(17)(16)....(4)(3)(2)(1)
So, we can move the 17 and the 3 in front to say that 50! = (17)(3)(50)(49)(48)(47)...
Since (17)(3) = 51, we can say that 50! = (51)(50)(49)(48)(47)...
In other words, 51 must be a factor (divisor) of 50!
We're looking for a number that is not a factor of 50!
So, let's keep looking.


50! = (50)(49)(48)(47)....(26)(25)....(4)(3)(2)(1)
So, we can move the 26 and the 2 in front to say that 50! = (26)(2)(50)(49)(48)(47)...
Since (26)(2) = 52, we can say that 50! = (52)(50)(49)(48)(47)...
In other words, 52 must be a factor (divisor) of 50!
We're looking for a number that is not a factor of 50!
So, let's keep looking.

What about 52?
50! = (50)(49)(48)(47)....(26)(25)....(4)(3)(2)(1)
So, we can move the 26 and the 2 in front to say that 50! = (26)(2)(50)(49)(48)(47)...
Since (26)(2) = 52, we can say that 50! = (52)(50)(49)(48)(47)...
In other words, 52 must be a factor (divisor) of 50!
We're looking for a number that is not a factor of 50!
So, let's keep looking.

What about 53?
Since 53 is prime, we cannot rewrite it as (something)(something).
In other words, we will not find 53 "hiding" within the product (50)(49)(48)(47)...
So, 53 is the smallest number that is not a factor of 50!
Unfortunately, the question asks us to find the smallest positive integer that is not prime.
So, if we take 53 and double it, we will have the smallest possible non-prime number that satisfies the given condition.

Finally, the question asks us to find the sum of the factors of x.

Posted Sat Aug 24, 2013 9:03 am
ganeshrkamath wrote:
Every positive integer less than or equal to 50 is a factor of 50!
The next composite number is 51.

Factors of 51: 1,3,17,51
Sum of the factors of 51 = 1+3+17+51 = 72

Choose C

Cheers
Be careful - 51 is a factor of 50!
51 = (3)(17)
50! = (50)(49)(48)(47)....(17)(16)....(4)(3)(2)(1)

On the other hand, we know that the prime number 53 cannot be a factor of 50!, since there is no way to "create" 53 with any of the numbers from the product 50!
So, (53)(2) will be the smallest non-prime integer that is not a factor of 50!

(53)(2) = 106
The factors of 106 are 1, 2, 53 and 106
Their sum = 1 + 2 + 53 + 106 = [spoiler]162 = D[/spoiler]

Cheers,
Brent
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by Azpire » Sun Apr 20, 2014 6:21 am
Brent@GMATPrepNow wrote:
melguy wrote:I am confused with the language of the problem. What is the key takeaway from this problem and how can I apply it on similar problems e.g. if we used 12! or 20! instead of 50!

In the case of 12! will the value be 13 x 2 = 26
In the case of 20! will the value be 23 x 2 = 46

Please help me with the problem. Thanks
Looks good, melguy.

If x is the smallest positive non-prime integer that is not a factor of k!, then x = (the smallest prime that's greater than k)(2)

Some examples
- The smallest positive non-prime integer that is not a factor of 14! is (17)(2)
- The smallest positive non-prime integer that is not a factor of 60! is (61)(2)
- The smallest positive non-prime integer that is not a factor of 50! is (53)(2).

Cheers,
Brent
Hi, if the above is true then if we consider 5! then the smallest prime after it is 7 and ttherefore the smallest non prime positive integer which is not a factor should have been 7*2=14. But the smallest non prime positive integer that is not its factor is 9.
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by Brent@GMATPrepNow » Sun Apr 20, 2014 7:41 am
Azpire wrote: Hi, if the above is true then if we consider 5! then the smallest prime after it is 7 and ttherefore the smallest non prime positive integer which is not a factor should have been 7*2=14. But the smallest non prime positive integer that is not its factor is 9.
You're absolutely right. I was too eager to create a general rule.
I believe the rule should have a few provisos, as follows:
If x is the smallest positive non-prime integer that is not a factor of k!, then x = (the smallest prime that's greater than k)(2) AS LONG AS k > 6
If k = 1, then 4 is smallest positive non-prime integer that is not a factor of k!
If k = 2, then 4 is smallest positive non-prime integer that is not a factor of k!
If k = 3, then 4 is smallest positive non-prime integer that is not a factor of k!
If k = 4, then 9 is smallest positive non-prime integer that is not a factor of k!
If k = 5, then 9 is smallest positive non-prime integer that is not a factor of k!

I THINK that proviso fixes the general rule, but I could be wrong.
Anyone else want to weigh in on this?

I've edited my initial post accordingly

Cheers,
Brent
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by Matt@VeritasPrep » Mon Apr 21, 2014 10:36 am
As an aside, this question (which is one of ours @ Veritas Prep) is one of the harder ones in our CATs; I can't recall the exact numbers offhand, but the % correct is very low. Our fearless leader Brian Galvin discusses it here.
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