Discrete Math: Floor Functions

[rd_wingman]

here are a few questions I am having trouble with using the floor function. can anyone help and explain what's going on? thanks

1) Suppose n and d are integers and d is not 0. Prove the following:
a) If d|n, then n = floor(n/d)*d
b) If n = floor(n/d)*d, then d|n
c) Use the floor notation to state a necessary and sufficient condition for an integer n to be divisible by and integer d

2) For any real number x, if x is not an integer, then floor(x) + floor(-x) = -1

3) For all real numbers x, floor([floor(x/2)/2]) = floor(x/4)

 

[daon]

1) If d|n then n=d*k. Then [n/d]=[d*k/d]=[k]=k, so that [n/d]*d=k*d=n.

If [n/d]*d =n, then [n/d] must be an integer, say k (if it were not, neither would be n). Hence k*d=n => d|n.

Use these two to formulate an "if and only if" for part c.

For the rest, what properties of the reals and/or floor function can you use up to this point?

 

[daon]

I'm not sure if these are your definitions, but it works:

Assume x is not an integer.

[x] = n in Z such that n < x < n+1 => n > x-1
[-x] = m in Z such that m < -x < m+1 => m > -x-1

From these we see that [x]+[-x] = n+m < 0 AND n+m > -2.

So we have -2 < [x] + [-x] < 0. Since [x]+[-x] must be an integer, what must it be?