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Re: If the integer n is greater than 1, is n equal to 2?  [#permalink]

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dimitri92 wrote:
If the integer n is greater than 1, is n equal to 2?

(1) n has exactly two positive factors.
(2) The difference of any two distinct positive factors of n is odd.

Given: Integer n is greater than 1

Target question: Does n = 2?

Statement 1: n has exactly two positive factors.
In other words, statement 1 tells us that n is prime
There are several values of n that satisfy statement 1. Here are two:
Case a: n COULD equal 2, since 2 has exactly two positive factors: 1 and 2. In this case, the answer to the target question is YES, n equals 2
Case b: n COULD equal 3, since 3 has exactly two positive factors: 1 and 3. In this case, the answer to the target question is NO, n does NOT equal 2
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: The difference of any two distinct positive factors of n is odd.
Nice!!!!
Since n is greater than 1, we know that n has at least 2 factors.
We also know that 1 is a factor of ALL positive integers, AND we know that n is also a factor of n

Some important rules:
#1. ODD +/- ODD = EVEN
#2. ODD +/- EVEN = ODD
#3. EVEN +/- ODD = ODD
#4. EVEN +/- EVEN = EVEN

Statement 2 indirectly tells us than n - 1 must be ODD
Since 1 is ODD, Rule #3 tells us that n must be EVEN
If n is EVEN, then 2 is one of the factors of n.
So far we know two of the factors of n: 1 and 2

At this point, we can conclude that 1 and 2 are the ONLY factors of n (that is, n = 2)
How can we can we conclude this?

We already know that 1 (ODD) and 2 (EVEN) are factors of n.
If there existed another factor of n, that factor would have to be EVEN or ODD
If that factor were ODD, then the difference between that number and 1 (ODD) would be EVEN, and this betrays statement 2.
If that factor were EVEN, then the difference between that number and 2 (EVEN) would be EVEN, and this betrays statement 2.
So, we can be certain that 1 and 2 are the ONLY factors of n, which means n = 2.

The answer to the target question is YES, n equals 2
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Cheers,
Brent
_________________ Re: If the integer n is greater than 1, is n equal to 2?   [#permalink] 02 Aug 2019, 06:29

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