claudayst wrote:How many distinct prime divisors does a positive integer N have?
1) 2N has one prime divisor
2) 3N has one prime divisor
I understand from the S1 that N can only be 2 raised to the nth...same with S2 but with 3 raised to the nth..i answered each statement alone is sufficient which doesn't make sense i know since each statement should yield the same answer. Can someone explains why both statements together are needed? Thx in advance.
You're 100% correct when you say "from S1, N can only be 2 raised to some power." However, N can be N to the power of 0. In other words, N can equal 1.
Here's the full solution.
Target question:
How many distinct prime divisors does a positive integer N have?
A few words about prime divisors: A prime divisor of N is a prime number that is a divisor of N.
For example, since 12 = (2)(2)(3), then the distinct prime divisors of 12 are 2 and 3.
Similarly, since 30 = (2)(3)(5), then the distinct prime divisors of 30 are 2, 3 and 5.
Since 81 = (3)(3)(3)(3), then 3 is the only distinct prime divisor of 81.
Since 9 = (3)(3), then 3 is the only distinct prime divisor of 9.
etc.
Statement 1: 2N has one prime divisor
This tells us that 2 must the sole prime divisor of 2N.
However, 2N can be any power of 2 (2, 4, 8, 16, 32, etc.)
Now consider these two conflicting cases.
Case a: 2N = 2, which means N = 1, in which case
N has 0 distinct prime divisors
Case b: 2N = 4, which means N = 2, in which case
N has 1 distinct prime divisor
Since we cannot answer the
target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: 3N has one prime divisor
Using the same logic as above, we know that 3 must the sole prime divisor of 3N.
So, 3N can be any power of 3 (3, 9, 27, 81, etc.)
Now consider these two conflicting cases.
Case a: 3N = 3, which means N = 1, in which case
N has 0 distinct prime divisors
Case b: 3N = 9, which means N = 3, in which case
N has 1 distinct prime divisor
Since we cannot answer the
target question with certainty, statement 2 is NOT SUFFICIENT
Statements 1 and 2 combined:
For 2N
and 3N to have exactly 1 distinct prime divisor, it must be the case that N = 1, in which case
N has 0 distinct prime divisors
Since we can answer the
target question with certainty, the combined statements are SUFFICIENT
Answer =
C
Cheers,
Brent
