kvcpk wrote:Another example: take 9.
Number of factors = 3
1,3,9
can be written as
1*9 only.
Hence only 1 way.
Ops... you forgot 9 = 3*3
kvcpk wrote:
So it turnsout ot be number of factors/2 if number of factors is even
ALMOST right (more about it at the end) ...
kvcpk wrote:
and (number of factors-1)/2 if number of factors is odd.
No!! Let´s correct this: (number of factors + 1)/2 if number of
positive factors is odd. (But read till the end!!!!)
Example: number 9, then 9 = 3^2 therefore (2+1) = 3 factors, therefore three initial pairs:
(1, 9), (3,3) and (9,1) , so you subtract the (3,3) pair , divides by 2 (one correspondent for each "duplicity") and put back the (3,3) you subtracted, that is: (number of factors - 1) /2 + 1 = (number of factors+1)/2 , that´s the (almost) right answer!! (Check the equality I mentioned).
Final Explanation/Correction: the answer is
B, because negative factors were not excluded from the question stem! That means that kvcpk should just double his final argument, because for every par (a´,b´) of positive integers such that a.b = 6,084 you will have a different pair (a´, b´) of negative integers with the same property! (a´ and b´ are the opposites of a and b, respectively.)
Obs.: please read my other post (below) to understand why the answer is not
E... at this post we did not excluded equal factors...
Example 01: let´s do it for the number 12... (not a perfect square)
(1, 12) --> (-1, -12)
(2, 6) --> (-2, -6)
(3, 4) -- (-3, -4)
Using our formula: 6/2 = 3 and then double it to get 2.3 = 6, correct!!
Example 02: let´s do it for the number 25... (a perfect square)
(1, 25) --> (-1, -25)
(5, 5) --> (-5, -5)
Using our formula: (3+1)/2 = 2 and then double it to get 2.2 = 4, correct!!
Important: in this problem, we must take out same factors (question stem), therefore AT THE END (of case/example 02, that is, when we have a perfect square) we subtract always 2, that is (a, a) and (-a, -a) where a is the arithmetic root of the (perfect square) number given... more on this matter in my next post, right after Rahul´s...
Regards,
Fabio.
