In Analysis 1 in chapter two section 3 on multiplication one of the exercises is the proof of the following.
(Euclidean algorithm). Let n be a natural number, and let q be a positive number. Then there exist natural numbers m, r such that 0≤r < q and n = mq +r.
With the hint to induct on n. I am having trouble navigating this proof with only without the idea of subtraction and only cancellation laws for addition and multiplication.
A proof would be helpful for me to reference.