[原创]关于2-HEIGHT 题

10、For all x, x is positive integer,  "2-height" is defined
    to be the greatest nonnegative n of x, what is the
    greatest number of 2-height when 2" is the factor of x?
A. 2
B. 12
C. 40
D. 76
E. 90

Answer: (by Anchoret)
A. 2=2^1
B. 12=2^2*3
C. 40=2^3*5
D. 76=19*2^2
E. 90=45*2^1
B is the answer.

1. X is a positive integer. 2-height of X is defined as the greatest negative integer n where 2^n is a factor of X. K and M are two positive integers. Whether 2-height of K is greater than 2-height of M?
a. K is greater than M
b. K is even times of M
(Key: B)

(by rosemsem)
题义解析：说对于含2的n次方的数, 2-height 指的是n的值。问k和m谁的2-height大？
(1) K>M
(2) K除以M是偶数.
(please notice K,k; M,m; e)

K = a* 2^k;
M = b* 2^m;

(1) k>m, means nothing.
(2) k/m= (a/b) * (2^k/2^m) = 2^e;
A, b must be odd number, or you can extract at least one more 2, which gonna change k or m. So in this case, (a/b) must be 1, otherwise it would be a fraction.
In a word, k-m=e. K>m.

B is sufficient.