luisnavarro wrote:
Hi,
Explanation seems good, but it look hard for me; could you develop with more examples or detail the answer?
Thanks a lot.
Luis Navarro
Looking for 700
If for any positive integer x, d[x] denotes its smallest odd divisor and D[x] denotes its largest odd divisor, is x even?First of all note that the smallest positive odd divisor of any positive integer is 1. Thus \(d[x]=1\) for any x.
(1) D[x] - d[x] = 0 --> \(D[x] - 1 = 0\) --> \(D[x] = 1\) --> x can be 1, so odd or \(2^n\), (2, 4, 8, ...), so even. Not sufficient.
(2) D[3x] = 3 --> again x can be 1, so odd, as the largest odd divisor of \(3x=3\) is 3 or x can be \(2^n\) (2, 4, 8, ...), so even, as the largest odd divisor of 3*2=6 or 2*4=12 is 3. Not sufficient.
(1)+(2) From (1) and (2) we have that x can be either 1, so odd or 2^n, so even. Not sufficient.
Answer: E.
Very interesting, great¡¡¡ Actually in my analysis I had not view the 2^n posibilities, and now it is clear to me: For all 2^n the smallest and greatest odd divisor are the same = 1. It is easy to understand when you actually see explanations like yours but, its hard to do "alone".
Thanks.