If S is the infinite sequence S1=9, S2=99, S3=999..., SK = 10^K-1,..., is every term in S divisible by the prime number p?

1) p is greater than 2.

2) At least one term in sequence S is divisible by p.

In the MGMAT Number Properties book, the rephasing of the question becomes: Is p = 3?
How did it arrive at that?

I also read the answer explanation in the OG book but still find it confusing.

Is there any dumbed down explanation of what's really going on in this problem?

is the problem supposed to read
SK = (10^K)-1
or
SK = 10^(K-1)
?

_________________
Stacey Koprince
Instructor
Director of Online Community
ManhattanGMAT

Sk=(10^k)-1

Is the answer E?

If S is the infinite sequence S1=9, S2=99, S3=999..., SK = 10^K-1,..., is every term in S divisible by the prime number p?

1) p is greater than 2.

2) At least one term in sequence S is divisible by p.

Looking at 1 in isolation - In this case, p could be 3 or 5 (or 7, 11, etc.) - using 3, we get a "yes" and using 5, a "no," so insufficient.
Looking at 2 in isolation - This implies p could be 11 (99 is divisible by 11, while 999 isn't - so "no") or 3 (all terms divisible by 3, so "yes") - insufficient because we have attained conflicting answers.
Using 1 and 2 together doesn't allow us to attain a concrete answer either (p could still be 3 or 11...), so answer must be E, I think...

Actually i would think it's E also. If At least one term in sequence S is divisible by p onloy means that it can be either S1 or S9 or S23, whatever. S23 is not divizible to S3, becasue S3 is a smaller number, hence giving a fraction. I personally would go with E. Official answer please?
Thanks,

Answer is E, yes.

The initial rephrasing is just based on the thought "is there a prime number for which I could always get a definitive answer?"

Given the pattern 9, 99, 999, 9999, etc. let's try some small prime numbers and see what we can figure out.

If p is 2, then I'd have a definitive answer (no, every term is NOT divisible by p).

What about 3? The divisibility rule for 3 states that the digits have to add up to a multiple of 3. If the digits are always 9's that would work.

What about 5? Definitive answer no because the numbers in the sequence will never end in 0 or 5. I don't want to go forever with this - I just note that I need to figure out the value of prime number p in order to answer the question because I'm getting conflicting answers depending on p's value.

I'm not a huge fan of rephrasing this one "is p=3?" because I think the question is a bit more complex. If, for example, you could determine that p=2, then you could answer this definitively no (but statement 1 kills that possibility). I'd rather rephrase this as "what is p?"

(1) p > 2. This still leaves 3 and 5 as possibilities and I already decided above that 3 and 5 give different answers, so this is insufficient.

(2) at least one term is div. by p. (but not necessarily all of them!). Well, given my above numbers (2, 3, and 5), only 3 works. But there are other prime numbers, so let's try something else just to make sure. How do I know what to try? Use the sequence. The first number in the sequence is 9. Well, I've already dealt with that one b/c I already know 3 works and 3 is the only prime factor of 9. The second number is 99. 99 = 9*11 = 3*3*11. I already know 3 works so ignore it. Now I also know that 11 is a prime divisor of one term (99). Is it a divisor of ALL terms? No - it's not a divisor of 9, for example. Insufficient.

(1) AND (2) What's my overlap? (1) leaves open every possible prime except for 2. So 3 and 11 are possibilities. (2) also allows 3 and 11. 3 gives a "yes" answer to the question and 11 gives a "no" answer to the question... so I can't tell. Insufficient.

_________________
Stacey Koprince
Instructor
Director of Online Community
ManhattanGMAT