Elementary Number Theory and Its Application, 6th Edition by Kenneth H. Rosen Exercises 1.1.20

20. Show that if m is a positive integer,then [mx] = [x] + [x + (1/m) ] + [x + ( 2/m) ] + · · ·+ [x + (m - 1) /m] whenever x is a real number.

We have to show that m if is a positive integer, then
Whenever x is a real number
We consider
Where [x] is the greatest integer part of x and {x} is the fractional part of x
We also know that
We now, consider the right hand side of (1)
Where p is the minimum value such that
Proceeding further, we get,
RHS=m[x]+0+0+0+...p times+[{x}+p/m]+...
=m[x]+1+1+1+...(m-p) times
Now, since,
Implies that
Which in turn implies that
{x}=(m-p)/m+(<1/m) for p to be minimum
[m{x}]=m-p …. (2)

Therefore, by (2) we get

Hence, we have prove (1)

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