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

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 ?

• is that a question on notation specifically? if so, add notation tag – gt6989b Jan 17 at 15:32
• @gt6988b yes but I can't edit my post. – Profahk Jan 17 at 15:38

$$\ 16 = 5\times 3 + 1$$ 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.

• Thank You.Do you have any reference (book, website,...) that I can convince my colleagues ? – Profahk Jan 17 at 17:43

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.

• But we can write an equality to express an Euclidean division ? – Profahk Jan 17 at 15:41
• Both equalities $16=3\times5+1$ and $16=5\times3+1$ explain why is it that the quotient is $5$ and the remainder is $1$. But none of them is the Euclidean division. – José Carlos Santos Jan 17 at 15:43
• I agree that both of them explain why the quotient is 5 and the remainder 1. – Profahk Jan 17 at 15:46
• I think I'm misunderstood. – Profahk Jan 17 at 15:47