tonebeeze wrote:Can someone please walk me through the logic in statement 2.
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.
Hi,
first off, it's important to understand exactly what (2) says - I've seen many people get this question wrong because they misinterpret (2).
The difference of any two distinct positive factors of n is odd.
This means that if you take any two distinct factors of n, you'll get an odd difference. How is this possible? Only in 1 case: n must have exactly 1 even factor and exactly 1 odd factor.
Let's examine why this conclusion holds:
if n had two odd factors, then if we took the difference between those two factors we'd get an even result and violate the condition;
if n had two even factors, then if we took the difference between those two factors we'd get an even result and violate the condition; and
if n has more than two factors, then we'll always have at least two evens or at least two odds in the mix, leading to a violation of the condition.
Since 2 is the only number that has exactly 1 even factor and exactly 1 odd factor, (2) gives us a definite "yes" answer and is sufficient.
