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(1) 2N has one prime divisor => N = 1 or N = 2 => N can have either 1 or 0 prime factor =>insuff (2) 3N has one prime divisor => N = 1 or N = 3 => N can have either 1 or 0 prime factor => insuff

(1)& (2) => N = 1 => N has no prime factor => suff hence C

The answer is C - you need both pieces of information to know for sure how many prime divisors N has.

My explanation

So for 2N to have 1 prime divisor, then N cannot be any prime number but 2 (because for e.g. 2 * 3 = 6, and 6 has two prime divisors - 2 and 3), nor can it be a composite number that can be broken down into any other prime numbers but 2 (for e.g. 18 can be broken down into 3 * 3 * 2, but 16, can only be broken down into 2 * 2 * 2 *2). Therefore, it doesn't matter what N is, as long as it satisfies these assumptions. So N can be 1, 2, 4, 8, etc... THEREFORE: If N is 1, then N has no prime divisors. But if N is 2, or 4, or 8, etc... then N has 1 prime divisor, which is 2. So this is insufficient by itself.

The same applies for 3N having 1 prime divisor, such that, N can be any number as long 3 is the only prime divisor. As such, N can be 1, 3, 9, 27, etc... If N is 1, then it has 0 prime divisors, and if N is 3, or 9, or 27, etc... then N has 1 prime divisor, which is 3. So this is insufficient by itself.

However, for 2N AND 3N to only have 1 prime divisor, then N must be equal to 1. Thus, we know N has no prime divisors, because 1 has no prime divisors. Therefore, Both of them together are sufficient.

If 2N has only 1 prime divisor, it could have only been that either: N has no prime divisor or N has only digit 2 as a prime divisor/factor.... 2, 4, 8...(i.e 2^n) INSUFFICIENT

By the same toke, either N has no prime divisor or N has only digit 3 as a prime divisor/factor....3, 9, 27...(i.e 3^n) INSUFFICIENT

Combining (1) & (2) => 1,2 or 1,3 only digit "1" is common...N has no distinct prime divisor. Thus, the correct response is C
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If 2N has only 1 prime divisor, it could have only been that either: N has no prime divisor or N has only digit 2 as a prime divisor/factor.... 2, 4, 8...(i.e 2^n) INSUFFICIENT

By the same toke, either N has no prime divisor or N has only digit 3 as a prime divisor/factor....3, 9, 27...(i.e 3^n) INSUFFICIENT

Combining (1) & (2) => 1,2 or 1,3 only digit "1" is common...N has no distinct prime divisor. Thus, the correct response is C

I thought the answer was B, since A had 2*2=4 and 2*1=2 with 2 being the only prime divisor. Only to realize that 3*3 = 9 which also has 2 divisors and therefore cannot be the answer.
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The answer is C - you need both pieces of information to know for sure how many prime divisors N has.

My explanation

So for 2N to have 1 prime divisor, then N cannot be any prime number but 2 (because for e.g. 2 * 3 = 6, and 6 has two prime divisors - 2 and 3), nor can it be a composite number that can be broken down into any other prime numbers but 2 (for e.g. 18 can be broken down into 3 * 3 * 2, but 16, can only be broken down into 2 * 2 * 2 *2). Therefore, it doesn't matter what N is, as long as it satisfies these assumptions. So N can be 1, 2, 4, 8, etc... THEREFORE: If N is 1, then N has no prime divisors. But if N is 2, or 4, or 8, etc... then N has 1 prime divisor, which is 2. So this is insufficient by itself.

The same applies for 3N having 1 prime divisor, such that, N can be any number as long 3 is the only prime divisor. As such, N can be 1, 3, 9, 27, etc... If N is 1, then it has 0 prime divisors, and if N is 3, or 9, or 27, etc... then N has 1 prime divisor, which is 3. So this is insufficient by itself.

However, for 2N AND 3N to only have 1 prime divisor, then N must be equal to 1. Thus, we know N has no prime divisors, because 1 has no prime divisors. Therefore, Both of them together are sufficient.

How many distinct prime divisors does a positive integer N have? Rephrase: What is N?

S1: 2N has one prime divisor Given 2N has one prime divisor. The prime divisor should be the prime 2. However, 2N may have one or more non-prime divisors maintaining the fact that 2 is the only prime divisor. So, 2N may be equal to 2, 4, or multiples of 2. Thus, N = 1, 2, or other multiples of 2. So, S1 is not sufficient. Eliminate AD.

S2: 3N has one prime divisor Given 3N has one prime divisor. The prime divisor should be the prime 3. However, 3N may have one or more non-prime divisors maintaining the fact that 3 is the only prime divisor. So, 3N may be equal to 3, 9, or multiples of 3. Thus, N = 1, 3, or other multiples of 3. So, S2 is not sufficient. Eliminate B.

Both S1 & S2: N = 1 is common in values of S1 and S2. Sufficient to answer that N = 1.

At the outset you should ask questions: What is N?

In first statement, 2N consists of one prime divisor. Clearly that prime divisor is 2. So, what are the possible values of N? N can be equal to 1 or 2. (2N) will have one prime divisor only in these two situations. So, N = 1 or 2, which is not sufficient.

Similarly, in second statement, 3N consists of one prime divisor. Clearly that prime divisor is 3. N can be equal to 1 or 3 in this case. (3N) will have one prime divisor (3) only if N = 1 or 3, which is not sufficient.

Both statements: give N = 1 which has no prime divisor. But, what's important is that it gives one clear answer. So, Sufficient. C is correct.

In first statement, 2N consists of one prime divisor. Clearly that prime divisor is 2. So, what are the possible values of N? N can be equal to 1 or 2. (2N) will have one prime divisor only in these two situations. So, N = 1 or 2, which is not sufficient.

My doubt :: The statement 1 states that 2N has one prime divisor . It does not tell that 2N has one distinct prime divisor Now taking this into consideration my doubt is even if N=2 => 2N is 4 which means there are 2 prime divisors i.e both the 2's. So if statement 1 has to be true I think 2N=2 and so N has to be 1 and so we can solve the question.

I may be wrong , but just trying to understand...please help...

@prakarp: I think you are very close. Here is how I'd approach this:

Statement 1 says: (2N) has one prime divisor. If (2N) has one prime divisor and the prime divisor is 2, then 2 is the distinct prime divisor in this case. However, what's important is the fact that there are no other prime divisors in N. So, possible values of (2N) are 2, 4, 8, and other multiples of 2 ensuring that the distinct prime divisor remains 2. This leaves us with N = 1, 2, 4, and mutiples of 2. Not sufficient.

Applying the similar concept to statement 2, we will get N = 1, 3, 9 and multiples 3. Not Sufficient.

Combining both statements gets us to N equals 1 which has no prime divisor. Sufficient.

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I did not understand the QA, or rather i am not satisfied with it. Pls help me with the explanation .

Thanks.

BELOW IS REVISED VERSION OF THIS QUESTION:

How many distinct prime divisors does a positive integer \(n\) have?

(1) \(2n\) has one distinct prime divisor --> obviously that only prime divisor of \(2n\) is 2. So, \(2n\) can be 2, 4, 8, ... Which means that \(n\) can be 1, 2, 4, ... If \(n=1\) then it has no prime divisor but if \(n\) is any other value (2, 4, ...) then it has one prime divisor: 2 itself. Not sufficient.

(2) \(3n\) has one distinct prime divisor. Basically the same here: the only prime divisor of \(3n\) must be 3. So, \(3n\) can be 3, 9, 27, ... Which means that \(n\) can be 1, 3, 9, ... If \(n=1\) then it has no prime divisor but if \(n\) is any other value (3, 9, ...) then it has one prime divisor: 3 itself. Not sufficient.

(1)+(2) From above the only possible value of \(n\) is 1, and 1 has no prime divisor. Sufficient.

The answer is C - you need both pieces of information to know for sure how many prime divisors N has.

My explanation

So for 2N to have 1 prime divisor, then N cannot be any prime number but 2 (because for e.g. 2 * 3 = 6, and 6 has two prime divisors - 2 and 3), nor can it be a composite number that can be broken down into any other prime numbers but 2 (for e.g. 18 can be broken down into 3 * 3 * 2, but 16, can only be broken down into 2 * 2 * 2 *2). Therefore, it doesn't matter what N is, as long as it satisfies these assumptions. So N can be 1, 2, 4, 8, etc... THEREFORE: If N is 1, then N has no prime divisors. But if N is 2, or 4, or 8, etc... then N has 1 prime divisor, which is 2. So this is insufficient by itself.

The same applies for 3N having 1 prime divisor, such that, N can be any number as long 3 is the only prime divisor. As such, N can be 1, 3, 9, 27, etc... If N is 1, then it has 0 prime divisors, and if N is 3, or 9, or 27, etc... then N has 1 prime divisor, which is 3. So this is insufficient by itself.

However, for 2N AND 3N to only have 1 prime divisor, then N must be equal to 1. Thus, we know N has no prime divisors, because 1 has no prime divisors. Therefore, Both of them together are sufficient.

Hope this helps!

but statement 1 and 2 says : 2N has one prime divisor( not distinct) 3N has one prime divisor(not distinct)

The answer is C - you need both pieces of information to know for sure how many prime divisors N has.

My explanation

So for 2N to have 1 prime divisor, then N cannot be any prime number but 2 (because for e.g. 2 * 3 = 6, and 6 has two prime divisors - 2 and 3), nor can it be a composite number that can be broken down into any other prime numbers but 2 (for e.g. 18 can be broken down into 3 * 3 * 2, but 16, can only be broken down into 2 * 2 * 2 *2). Therefore, it doesn't matter what N is, as long as it satisfies these assumptions. So N can be 1, 2, 4, 8, etc... THEREFORE: If N is 1, then N has no prime divisors. But if N is 2, or 4, or 8, etc... then N has 1 prime divisor, which is 2. So this is insufficient by itself.

The same applies for 3N having 1 prime divisor, such that, N can be any number as long 3 is the only prime divisor. As such, N can be 1, 3, 9, 27, etc... If N is 1, then it has 0 prime divisors, and if N is 3, or 9, or 27, etc... then N has 1 prime divisor, which is 3. So this is insufficient by itself.

However, for 2N AND 3N to only have 1 prime divisor, then N must be equal to 1. Thus, we know N has no prime divisors, because 1 has no prime divisors. Therefore, Both of them together are sufficient.

Hope this helps!

but statement 1 and 2 says : 2N has one prime divisor( not distinct) 3N has one prime divisor(not distinct)

I think the revision is necessary as without it I took statement 1 to mean N has only 1 prime divisor, as in quantity, i.e. one 2, therefore N must be 1. Sufficient. Same for statement 2. With the addition of "distinct" to the two statements it is a good question.

I think the revision is necessary as without it I took statement 1 to mean N has only 1 prime divisor, as in quantity, i.e. one 2, therefore N must be 1. Sufficient. Same for statement 2. With the addition of "distinct" to the two statements it is a good question.

I did the same thing. Without "distinct", the statement infers that there is one, and only one prime divisor, making both statements sufficient. It's interesting how one word can make such a difference.