OG - DS - #132

[Pooja.goradia]

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.

[rajesh]

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

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.

[ayang]

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

[Guest]

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!

[Nov1907]

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

[StaceyKoprince]

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.