Euclidean division algorithm

While studying number theory, we will be primarily interested in the properties of the integers and natural numbers. This means that if we want to write something like "[math]a/b [/math]", we'll have to be very careful to check that [math]b [/math] divides [math]a [/math]. This makes everything very complicated, so instead, we'll avoid dividing natural numbers.

We can, however, do division with remainder. The Euclidean division algorithm is just a fancy way of saying this:

Claim (see proof):

For all [math]a \href{/cs2800/wiki/index.php/%E2%88%88}{∈} \href{/cs2800/wiki/index.php/%E2%84%95}{ℕ} [/math] and all [math]b \gt 0 [/math], there exists numbers [math]q [/math] and [math]r \href{/cs2800/wiki/index.php/%E2%88%88}{∈} \href{/cs2800/wiki/index.php/%E2%84%95}{ℕ} [/math] such that

1. [math]a = qb + r [/math]
2. [math]0 \leq r \lt b [/math]

Here [math]q [/math] and [math]r [/math] are the quotient and remainder of [math]a [/math] over [math]b [/math]:

Definition: Quotient

We say [math]q \href{/cs2800/wiki/index.php/%5Cin}{\in} \href{/cs2800/wiki/index.php/%E2%84%A4}{ℤ} [/math] is a quotient of [math]a [/math] over [math]b [/math] if [math]a = qb + r [/math] for some [math]r \href{/cs2800/wiki/index.php/%5Cin}{\in} \href{/cs2800/wiki/index.php/%E2%84%A4}{ℤ} [/math] with [math]0 \leq r \lt b [/math]. We write [math]q = quot(a,b) [/math] (note that quot is a well defined function).

Definition: Remainder

We say [math]r \href{/cs2800/wiki/index.php/%5Cin}{\in} \href{/cs2800/wiki/index.php/%E2%84%A4}{ℤ} [/math] is a remainder of [math]a [/math] over [math]b [/math] if [math]a = qb + r [/math] for some [math]q \href{/cs2800/wiki/index.php/%5Cin}{\in} \href{/cs2800/wiki/index.php/%E2%84%A4}{ℤ} [/math] and [math]0 \leq r \lt b [/math]. We write [math]r = rem(a,b) [/math] (note that rem is a well defined function).

Note that all this is a theorem, it is called the "Euclidean division algorithm" because its proof contains an algorithm.

Proof:

We prove this by weak induction on [math]a [/math]. Let [math]\href{/cs2800/wiki/index.php?title=P(n)&action=edit&redlink=1}{P(a)} [/math] be the statement "for all [math]b \gt 0 [/math], there exists [math]q, r \href{/cs2800/wiki/index.php/%E2%88%88}{∈} ℕ [/math] satisfying (1) and (2) above." We will show [math]P(0) [/math] and [math]P(a+1) [/math] assuming [math]P(a) [/math].

Base case: We want to show P(0). Let [math]q = r = 0 [/math]. Then [math]qb + r = 0b + 0 = 0 = a [/math] and [math]0 \leq r \lt b [/math] since [math]r = 0 [/math].

Inductive step: Assume [math]P(a) [/math]; we want to show [math]P(a+1) [/math]. By [math]P(a) [/math], we know that there exists numbers [math]q' [/math] and [math]r' [/math] such that [math]a = q'b + r' [/math]. We want to show that there exists numbers [math]q [/math] and [math]r [/math] such that [math]a + 1 = qb + r [/math].

Since [math]r' \lt b [/math], we know that either [math]r' + 1 \lt b [/math] or [math]r' + 1 = b [/math]. In the former case, we can let [math]q = q' [/math] and [math]r = r' + 1 [/math]. Then [math]qb + r = q'b + r' + 1 = a + 1 [/math] (and clearly [math]0 \leq r \lt b [/math]).

In the latter case, we can let [math]q = q' + 1 [/math] and [math]r = 0 [/math]. Then [math]qb + r = (q' + 1)b + 0 = q'b + (r' + 1) = a + 1 [/math]. Again it is clear that [math]0 \leq r \lt b [/math].

In either case, we have shown that there exist [math]q [/math] and [math]r [/math] satisfying (1) and (2), as required.

Note that even though this is a proof, it also contains an algorithm for finding the quotient and remainder. If [math]a = 0 [/math], follow the instructions in the base case; if [math]a \gt 0 [/math], then first find the quotient q' and remainder r' for [math]a' = a-1 [/math] and [math]b [/math], then follow the instructions in the inductive step. Of course, while we are finding [math]q' [/math] and [math]r' [/math], we may need to also find the quotient [math]q' ' [/math] and remainder [math]r' ' [/math] of [math]a' ' = a'-1 [/math] and [math]b [/math], and so on.

This is a general pattern: most proofs that something exists give you instructions for finding it. In fact, there are whole programming languages where instead of writing a program, you write a proof; after the computer checks your proof, it extracts a program for you (which is guaranteed to be correct!).