Find the remainder when N is divided by 1000. (Here ⌊x⌋ denotes the greatest integer that is less than or equal to x, and ⌈x⌉ denotes the least integer that is greater than or equal to x.)
Solution:
First note that
⌈x⌉−⌊x⌋={1, if x is not an integer 0, if x is an integer
Thus for any positive integer k,
⌈log2k⌉−⌊log2k⌋={1, if k not an integer power of 20, if k an integer power of 2.
The integers k,1≤k≤1000, that are integer powers of 2 are described by k=2j,0≤j≤9. Thus