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 factors of N is odd.
III. The number of distinct prime factors of N is even. Answers
A) I only
B) II only
C) I and II
D) I and III
E) I, II and III
SOLN:   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.   ANSWER:I and II
<result>C</result>
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 factors of N is odd.
III. The number of distinct prime factors of N is even.