Euclidean division? ( $16=5\cdot 3+1$ vs $16=3\cdot 5+1$)

Question

Is the equality $16=5\cdot 3+1$ the euclidean division of $16$ by $3$ or not ?

This question is a point of discord between teachers where some them state that the divisor must be written in the first position (in this example, one has to write $16=3\cdot 5+1$ i.e must write the divisor first then the quotient).

What do you think ?

Answer 1

The answer is neither yes nor no. “Euclidean division” is a concept, not an arithmetic expression. The quotient of the Euclidean division of $16$ by $5$ is $3$ and the remainder is $1$. You can prove it by any of the equalities $16=3\times5+1$ or $16=5\times3+1$ (they are equivalent, of course), but, again, none of them is the Euclidean division.

Answer 2

$\ 16 = 5\times 3 + 1$ could arise from dividing $16$ by $5$ or by $3$. Without any further context there is no way to determine if $5$ or $3$ is the intended divisor.

Though - as you mention - one could impose syntactic conventions that imply which is the intended divisor, this is usually not a good idea, since it violates referential transparency, i.e. replacing an expression with an equivalent expression should not change its meaning. But if we replace $3^2$ by $3\times 3$ in $\,37 = 3^2\times 4 + 1$ then it changes the divisor from $9$ to $3$ (using your convention that the first factor is the divisor).

Further, conventions that force one to use specific commutations / associations of products may lead to increased complexity, e.g. instead of writing $\, f(x) = (x^2+x+1)^n g(x) + 1$ we would be forced to instead write $\,f(x) = (x^2+x+1)(x^2+x+1)^{n-1}g(x) + 1$ to denote that $\,x^2+x+1\,$ is the intended divisor.

As such, it is generally better to avoid such conventions and instead explicitly state the divisor.