I am having some difficulties with the following DS problem in the Official Guide -11th edition.

Problem #132:
If the integer n is greater than 1, is n equal to 2?

1) n has exactly two positive factors
2) The difference of any two distinct positive factors of n is odd.

The correct answer is B.

Here is what i do not understand: on statement 2, if you choose an even number greater than 2, such as 10 - what if the two distinct factors i chose to compare were 1 and 10, then the difference would be odd. In the explanation, it says if n>2, and n is even, then 2 and n are factors of n, and their difference is even. Thus, no integer greater than 2 satisfies this statement. However n=2 does satisfy this statment since 1 and 2 are the only positive factors of 1 and 2 and their difference is odd.

I understand what they are saying, but am having trouble seeing how they are testing the difference between the two "distinct" factors.

Thanks.

This means you take any two distinct factors, and the difference between them will be odd. So in your example, 10 has factors of 1, 2, 5, 10. 10 and 1 have odd difference but 10 and 2 don't. So our condition is not satisfied.

Statement (1) indicates, essentially, that n is prime.
Statement (2) indicates that the difference of any (read as "all") distinct positive factors of n is odd.

2 is the only number that any factor pair will have an odd difference (2 - 1).

For all prime numbers, 3, 5, 7, etc., the difference will be even.

For any other even number, the difference between 2 and itself will be even.

I hope that this is helpful. - Andrew

Hi, Andrew---

What about n=4? I thought that 4 would be a val for n that satisfied statement (2). I see from your post that you said"any" means "all", but I interpreted "any distinct positive factors" to mean all "eligible" factors of n.

4's factors:
1 and 4,
2 and 2

BUT I thought that 2 and 2 should be tossed out since they are not distinct factors of 4, leaving only 4 and 1 to be considered.

Can you please let me know where I'm going wrong?

Thanks!

For four the distinct factors are 1, 2 and 4 4-2 is still even.

When they say "distinct" factors they do NOT mean to "toss out" a number entirely if it's repeated. They just mean: don't use the repeats. So don't toss 2 out - just don't use 2 twice.