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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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    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.2012-08-30

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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".2012-08-30