If S is the infinite sequence S1=9 S2=99 S3=999 SK=10^k-1

No, B is not correct. It's straight E: if p=3 then every term in S is divisible by p but if p=11 then some terms in S are divisible by p (for example 99 and 9999) and some are not (for example 9 and 999). Not sufficient.

Answer: E.

hope it's clear.
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Actually question is asking , is every term in S divisible by the prime number p?

S1: p is greater than 2..Means wat? it means p cud b 3 ,5,7,11. If we say 3 then ans will be yes, Bt if we say 5 then we say no. Thats why Insufficient.

S2: at least one term is divisible by p. so ans wud b 3 or 11. 3 wud be divisible by every term of S, Bt 11 cud not be divisible by first term 9, and 999 etc.

Take both statement togather. still we cant give the ans, because p cud b 3 or 11.
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+1 E

Question tends to make you just go for 3 & you choose C but as Bunuel pointed out if you use 11 the assumption fails

Hello Bunuel

I have a doubt here.
Since we know after combining the 2 statements that the Prime number will NOT be 3 but any other Prime number that divides at least 1 number in the sequence.
So now we know for SURE that EVERY TERM IS NOT Divisible by a particular Prime number( Which the Questions asks). 11 will also not divide all terms but few only.

SO should the Answer not be "C" ?
Please correct me.

Thankyou

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Why cannot p be 3?
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Hi All,
Can any of the moderators amend this post to create a gap between the question and the query posted by morya003?

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Done. Thank you.
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The only prime that divides all terms in the sequence: 9, 99, 999,.. is 3. So the question is essentially asking: is p=3?
(1) p can be any prime number greater than 2, hence not sufficient.
(2) p can be 3 or 11, hence not sufficient.
Both taken together, again insufficient to identify p as 3.

Answer: E

boomtangboy summarizes aptly that the question tends to prey on the mistake that the test taker will consider 3 as the only possible value from statement (2).

Here we can choose 11 and 3 as test cases and we will get E answer fairly easily
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From question stem we know S={9,99,999,9999,99999,999999...................99999999999999....}
We can easily eye ball that this is divisible by either 3 or 11 or a combination of their multiples.

(1) p is greater than 2.
Insufficient :- p can be 3 or 5 or 7 or 11 or 13
In some cases p divides ; in some cases it don't

(2) At least one term in sequence S is divisible by p.
Insufficient again
starting from the second term (99) all terms are divisible by either 3 and 11
But the first team 9 is divisible only by 3 and not by 11

So we cannot for sure whether the prime p that the question stem is referring to is 3 or 11.

BOTH STATEMENT INSUFFICIENT
ANSWER IS E

Question is tricky and seems very tough in the first instance. Though I understood that every term is not divisible by p, but I marked C option due to my understanding that every term is not divisible by p (I did a mistake here... Every term is divisible by 3 and not by 11 as correctly suggested by @bunuel). So, the reading and understanding the language of the question in such problems is very important.

So, finally the solution goes like this.

Given : infinite sequence: S1=9, S2=99, S3=999, ..., SK = 10^K-1
DS: is S1, S2, S3, ......, SK divisible by the prime number p.

Statement 1 : P>2. So, p can be 3,5,7,11,.....
We can clearly see that every term is divisible by 3 but not by 5,7,11.....
NOT SUFFICIENT

Statement 2 : At least one term in sequence S is divisible by p. So, we can see that 9 is divisble by 3 and 99 is divisible by 11.
Here again every term is divisible by 3
But every term is not divisible by 11.
NOT SUFFICIENT

Combined : p>2 and At least one term in sequence S is divisible by p. So, p can be 3 or 11 or...
Its clear that every term is divisible by 3, but every term is not divisible by 11.
NOT SUFFICIENT

Answer E

but s5 is 9999 is clearly divisible by 11 right ?

Question: is every term in S divisible by the prime number p?

Statement 1:
If p = 3: Each term in S is divisible by the prime number 3 (3 is greater than 2)
If p = 11: Each term in S is NOT divisible by the prime number 11 (11 is greater than 2)

Thus, p = 3 or 11 => Hence, NOT Sufficient


Statement 2: At least one term in sequence S is divisible by p

If p = 3: Each term in S is divisible by the prime number 3 => At least one term in S is divisible by the prime number 3
If p = 11: The terms 99, 9999, etc. in S are divisible by the prime number 11 => At least one term in S is divisible by the prime number 11

Thus, p = 3 or 11 => Hence, NOT Sufficient

Combining: Even after combining, we have p = 3 or 11 => Hence, NOT Sufficient

Answer E
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