Bunuel wrote:
Orange08 wrote:
If the positive integer N is a perfect square, which of the following must be true?
I. The number of distinct factors of N is odd.
II. The sum of the distinct factors of N is odd.
III. The number of distinct prime factors of N is even.
For III, 1 is not considered as prime factor.
So, for example, for 4, distinct prime factor would be 2 only and not 2 and 1 both.
Likewise, for 9, distinct prime factor would be 3 only
Please write if my understanding is correct.
Yes, your understanding of III is right. Prime factor of 4 is 2 and prime factor of 9 is 3. So III is not alway true: a perfect square can have any number of prime factors.
Tips about the perfect square:
1. The
number of distinct factors of a perfect square is ALWAYS ODD.
2. The
sum of distinct factors of a perfect square is ALWAYS ODD.
3. A perfect square ALWAYS has an
ODD number of Odd-factors, and
EVEN number of Even-factors.
4. Perfect square always has
even powers of its prime factors.
So I and II must be true.
Let me try to explain it algebraically.
The perfect number A is factored as : a^2 x b^2 x c^2 (the power is always EVEN, for the sake of simplicity, I use the power of 2).
1. The # of distinct factors = 3 x 3 x 3 = 27 : ODD
2. The sum of distinct factors = (a^3 - 1) (b^3 - 1) (c^3 - 1) / [ (a-1)(b-1)(c-1) = (a^2 + a + 1)(b^2 + b + 1)(c^2 + c + 1)
because a,b,c are different primes then there is at most one even factor among a, b, and c. Let's say a = 2 -> (a^2 + a + 1) = EVEN + EVEN + ODD = ODD
b, c must be ODD -> (b^2 + b + 1) = ODD + ODD + ODD = ODD and (c^2 + c + 1) = ODD + ODD + ODD = ODD.
SO (a^2 + a + 1)(b^2 + b + 1)(c^2 + c + 1) = ODD x ODD x ODD = ODD.
3. A = a^2 x b^2 x c^2 if a is EVEN and b, c are ODD.
The # of possible factors of A = 3 x 3 x 3 = 27 = ODD.
The # of possible factors not containing a (so will be ODD) = 3 x 3 = 9 = ODD.
->
The # of possible factors containing a (so will be EVEN) is : 27 - 9 = 18 = ODD - ODD = EVEN.
Hope it helps.
A small addition to the above. Even if a is not even, (a^2 + a + 1) will result be odd. i.e. (ODD+ODD+ODD)=ODD