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Prime Number Q

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by DavidG@VeritasPrep » Sun Nov 22, 2015 9:56 am
dunnec3 wrote:If k is an integer greater than 1, and S is the sum of all positive divisors of k, is S > k + 1 ?

I can't see how you can rephrase 'Is S > k + 1 ?' to 'Is k not prime ?' Can anyone help me out?
Examine a few prime numbers to detect the trend. Say k = 3. If S is the sum of all the positive divisors of 3, then S = 1 + 3 = 4. Notice that this is equal to k + 1, or 3 + 1.

Say k = 5. If S is the sum of all positive divisors of 5, then S = 1 + 5 = 6. Again, S is equal to k + 1, or 5 + 1.

This makes sense. The only factors of a prime number are itself and 1. Therefore, if k is a prime number, the only factors of k will be 1 and k. The sum of all the factors would be 1 + k. If, by definition, the sum of the factors of a prime number 'k' is equal to k + 1, then any number whose sum is something other than k + 1 must be a non-prime number.
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by [email protected] » Mon Nov 23, 2015 10:14 am
Hi dunnec3,

It sounds like you're working on learning to 'rewrite' a DS question - that IS a useful skill (it can often make DS questions easier to handle and can save you time and effort), but it's not a necessary one to correctly answer DS questions.

David's suggestion to TEST VALUES to prove a pattern is useful in rewriting DS questions AND in solving them. Based on the DS prompt, I would expect that the two Facts would tell you something about K or S, which would allow you to TEST VALUES and answer the given question (whether you had rewritten it or not).

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by Matt@VeritasPrep » Fri Nov 27, 2015 1:14 am
Short answer: since 1 and k are always factors of k, the sum is always ≥ (k + 1) unless k itself is 1.

For instance,

1:: sum = 1
2:: sum = 1 + 2
3:: sum = 1 + 3
4:: sum = 1 + 2 + 4
5:: sum = 1 + 5
6:: sum = 1 + 2 + 3 + 6
7:: sum = 1 + 7
etc.

So you can see that the primes always give (# itself + 1) and the composites give something larger than that. This sort of pattern hunting is crucial on the GMAT: it helps shine light on so, so many intimidating and abstract problems.
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