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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
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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
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up vote 6 down vote accepted

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 $a=bq+r$.

The result is often called the Division Algorithm. Odd, because no algorithm is explicitly mentioned in the statement of the theorem.

Remark: Division Algorithm is the traditional name. Even Rosen's Elementary Number Theory (Fifth Edition), which is fairly computing-oriented, uses that name.

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I have known this as the "division theorem"; that name in Wikipedia is a redirect to the same article as "division algorithm". –  MJD Aug 30 '12 at 16:56
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