Data Sufficiency

Most of our students are trying to break the 700+ barrier. As a result, we've developed our own math problems written at the 700+ level; these are the types of questions you WANT to see, when you are working at that level. Try to solve this problem (before you peek at the answer).

Question
What is the positive integer n ?

(1) The sum of all of the positive factors of n that are less than n is equal to n
(2) n < 30

(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not.
(B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not.
(C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient to answer the question.
(E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.

Answer (highlight to read)

There is no conceptual or formulaic approach for solving this question. One must simply try out various integers.

(2) INSUFFICIENT: We can start with the second statement first because it is clear that it is insufficient to solve the question was is value of the positive integer n?

(1) INSUFFICIENT: We must first understand what this statement is saying. If all of n's factors (other than n itself) are added up, they equal n.

We can begin our search by considering prime factors. By definition prime factors have only two factors, themselves and 1. It is impossible that the factors "other-than-the number" add up to the number for any prime number. Thus we can begin our search for such n's with the number 4.

4 does not equal 1 + 2
6 DOES EQUAL 1 + 2 + 3
9 does not equal 1 + 3
10 does not equal 1 + 2 + 5
12 does not equal 1 + 2 + 3 + 4 + 6
14 does not equal 1 + 2 + 7
15 does not equal 1 + 3 + 5

At this point we might be tempted to think that this is a property that is unique to 6 and is unlikely to come around again (i.e. that the answer is A). It would behoove us to keep searching though and to at least cover the range defined by the second statement (i.e. n < 30) . If we do that we see that this property repeats itself one other time in the remaining integers that are less than 30.
16 does not equal 1 + 2 + 4 + 8
18 does not equal 1 + 2 + 9
20 does not equal 1 + 2 + 4 + 5 + 10
21 does not equal 1 + 3 + 7
22 does not equal 1 + 2 + 11
24 does not equal 1 + 2 + 3 + 4 + 6 + 8 + 12
25 does not equal 1 + 5
26 does not equal 1 + 2 + 13
27 does not equal 1 + 3 + 9
28 DOES EQUAL 1 + 2 + 4 + 7 + 14

The correct answer is E.