# Name of this division property

Let us take two integers, $a$ and $b$. Let us then take $\lfloor a / b\rfloor = c$ and $a \bmod b = d$. Obviously, it follows that $a = bc + d$. Our professor claimed that this was called the "Fundamental Theorem of Arithmetic" at his high school. Clearly, this isn't the case anymore, if it ever was.

What is this property actually called?

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For the benefit of other readers, $a/b=c$ is apparently referring to "integer division". Apparently programmers know about this convention but I don't think many mathematicians do. Personally I think this made-up ad-hoc pseudo-operation was invented to confuse and annoy mathematicians. (This is all written jocularly, of course!) – rschwieb Aug 30 '12 at 16:48
In some sense, the "division algorithm" theorem is the definition of the integer operations $a/b$ and $a%b$. – Thomas Andrews Aug 30 '12 at 16:52
Where are you located? The FTA has been pretty well-established as unique factorization for a long time. I knew a prof who said FTA should really be "If $a|bc$ and $a$ and $b$ are relative prime, then $a|c$." But he was an outlier. – Thomas Andrews Aug 30 '12 at 16:55

The result says that for any integers $a$ and $b$, with $b\gt 0$, there exist unique integers $q$ and $r$ such that $0\le r\lt b$ and $a=bq+r$.