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# M18-37

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Math Expert
Joined: 02 Sep 2009
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16 Sep 2014, 01:05
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Difficulty:

75% (hard)

Question Stats:

45% (01:28) correct 55% (01:19) wrong based on 192 sessions

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How many distinct prime divisors does a positive integer $$n$$ have?

(1) $$2n$$ has one distinct prime divisor

(2) $$3n$$ has one distinct prime divisor

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Joined: 02 Sep 2009
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16 Sep 2014, 01:05
Official Solution:

(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.

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19 Aug 2015, 15:16
We need to find PRIME divisor. Which means only prime number. If 2n has 1 prime divisor, thus n doesn't have any prime divisor. The same goes for 3n.
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09 Oct 2016, 11:16
Bunuel wrote:
Official Solution:

(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.

my doubt: we could interpret this from each statement alone that only 1 has no prime divisors & no other number satisfies this condition, then why go further?
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10 Oct 2016, 00:06
2
yashrakhiani wrote:
Bunuel wrote:
Official Solution:

(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.

my doubt: we could interpret this from each statement alone that only 1 has no prime divisors & no other number satisfies this condition, then why go further?

From each statement n can take more than one value. For example, for (1) n can be can be 1, 2, 4, ... For all these cases 2n has one distinct prime divisor, namely 2.
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11 Jul 2019, 14:30
Is this the same as saying
Statement 1: $$2n=2^n$$
Statement 2: $$3n=3^n$$?
Re: M18-37   [#permalink] 11 Jul 2019, 14:30
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# M18-37

Moderators: chetan2u, Bunuel