How can you derive an algorithm for dividing natural numbers?

In the set of natural numbers, division is defined as ${(a/b)*b=a}$

With the help of Peano’s axioms, I can find an algorithm for multiplication and addition. But how I can find an algorithm for the division operation which based on the axioms and laws of natural numbers?
For example, on Wikipedia, the quotient is found using repeated subtraction. But how can I be sure that this algorithm will always give the correct answer? On the basis of what laws are such calculation algorithms based?