# Identities or inequalities for $\lfloor$(a + b)/d$\rfloor$ (for integers a, b and d; d > 0)

I'm keen to know of any potentially useful identities or inequalities involving the expression $\,\lfloor$ (a + b) / d $\rfloor\,$, where $\;$a, b, d $\in$ Z; d $\gt$ 0.

From first principles (Euclidean division), all that I've been able to prove so far, is the following:

$\qquad$$\lfloor(a + b)/d\rfloor\,$$\;=\;$ $\,\lfloor$a/d$\rfloor\,$ + $\,\lfloor$b/d$\rfloor\,$ + c$\quad$(where c = 0 or c = 1)

So, an obvious inequality would be:

$\qquad$$\,\lfloor$a/d$\rfloor\,$ + $\,\lfloor$b/d$\rfloor\,$ $\;\le\;$ $\lfloor$(a + b)/d$\rfloor\,$ $\;\le\;$ $\,\lfloor$a/d$\rfloor\,$ + $\,\lfloor$b/d$\rfloor\,$ + 1

Are there any known potentially useful identities or inequalities for $\lfloor$(a + b)/d$\rfloor\,$ other than the above? (I couldn't find any after searching the web for just under an hour!)

• What does $a|d + b|d$ mean? What does $a|d = c|d$ mean? $a|d$ isn't a number or even a class of number. It is a statement about two numbers. What does "2 divides into 6 evenly plus 5 divides into 45 evenly" mean? what does "2 divides into 6 equals 5 divides into 10" mean? – fleablood Apr 26 '17 at 21:02
• @fleablood By a div d, what I mean is the quotient obtained upon dividing a by d (integer division). So, a = (a div d) d + (a mod d) , where a mod d is the remainder obtained in the aforementioned division. Apologies if the notation I'm using is unorthodox. – memexor Apr 26 '17 at 21:24
• Unorthodox is right! Google the floor function. – fleablood Apr 26 '17 at 21:27
• @fleablood Thanks for setting me right regarding the correct notation to use! I've edited my question accordingly. – memexor Apr 26 '17 at 21:53