Divisibility: If x is a positive integer, is x-1 a factor of

II wrote:If x is a positive integer, is x – 1 a factor of 104?

(1) x is divisible by 3.

(2) 27 is divisible by x.

Statement 2 gives us more direct info, so let's start there.

(2) 27 is divisible by x; in other words, x is a factor of 27.

So, x = 1, 3, 9 or 27

If we pick x=3, we get "is 2 a factor of 104?" yes!
If we pick x=9, we get "is 8 a factor of 104?" yes!
If we pick x=27, we get "is 26 a factor of 104?" yes!
If we pick x=1, we get "is 0 a factor of 104?" Doh! no!

[fun math fact: 0 is a multiple of every integer but a factor of no integer.]

So (2) is insufficient.

(1) x is divisible by 3; in other words, x is a multiple of 3.

If we pick x=3, we get "is 2 a factor of 104"? yes!

If we pick x=300, we get "is 299 a factor of 104?" no!

So (1) is insufficient.

When we combine (2) and (1), the only possible values for x are 3, 9 and 27. From our work with statement (2) we already know that all 3 of these give us "yes" answers to the original question. Combined, the statements are sufficient: choose (c).

you are the number properties master !

I thought the answer was B, and it never occurred to me that 0 would not be considered a factor of 26...

26 x 0 = 0 but 26/0 does not = 0, it's infinity/undefined.

Completely overlooked that. That sort of thing is why you do this for a living, and I don't![/img]