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? [/quote]
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DavidG@VeritasPrep
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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.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?
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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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).
GMAT assassins aren't born, they're made,
Rich
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).
GMAT assassins aren't born, they're made,
Rich
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Matt@VeritasPrep
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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.
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.
