th03 wrote:
Which of the following must be true if the square root of X is a positive integer?
I. X has an even number of distinct factors.
II. X has an odd number of distinct factors.
III. The sum of X’s distinct factors is odd.
(A) I only
(B) II only
(C) I and III
(D) II and III
(E) I, II, and III
Official answer is D.
I have a doubt with the answer for this question. I believe that the right answer should be A. If X=4, then its factors are 1,2,4,-1,-2,-4. Should't we consider negative factors too??? Please explain. Thanks!
Factor is a "positive divisor" (at least on the GMAT). So, the factors of 4 are 1, 2, and 4 ONLY.
Tips about perfect squares >0:
1. The
number of distinct factors of a perfect square is ALWAYS ODD.
The reverse is also true: if a number has the odd number of distinct factors then it's a perfect square;
2. The
sum of distinct factors of a perfect square is ALWAYS ODD.
The reverse is NOT always true: a number may have the odd sum of its distinct factors and not be a perfect square. For example: 2, 8, 18 or 50;
3. A perfect square ALWAYS has an
ODD number of Odd-factors, and
EVEN number of Even-factors.
The reverse is also true: if a number has an ODD number of Odd-factors, and EVEN number of Even-factors then it's a perfect square. For example: odd factors of 36 are 1, 3 and 9 (3 odd factor) and even factors are 2, 4, 6, 12, 18 and 36 (6 even factors);
4. Perfect square always has
even powers of its prime factors.
The reverse is also true: if a number has even powers of its prime factors then it's a perfect square. For example: \(36=2^2*3^2\), powers of prime factors 2 and 3 are even.
According to this, only II and III must be true.
Answer: D.
Hope it helps.
Thanks Bunuel, Could you please clarify the term "Distinct Factors"?