What is the rationale behind the below algorithm :
Algorithm to Convert From Decimal To Another Base
Let n be the decimal number.
Let m be the number, initially empty, that we are converting to. We'll be composing it right to left.
Let b be the base of the number we are converting to.
Repeat until n becomes 0
Divide n by b, letting the result be d and the remainder be r.
Write the remainder, r, as the leftmost digit of b.
Let d be the new value of n.
Statement: the remainder r is the last digit of the result.
Indeed, separate n into two parts: the part divisible by r and the remainder.
We get n = d * b + r where 0 <= r < b.
In base b, the notation is n = c_k * b^k + ... + c_1 * b^1 + c_0 * b^0.
Since everything except c_0 * b^0 is divisible by b, the equality r = c_0 follows trivially.
For the remaining digits, rewrite
n = c_k * b^k + ... + c_1 * b^1 + c_0 * b^0
as
n = (c_k * b^{k-1} + ... + c_1 * b^0) * b + c_0 * b^0.
On the other hand,
n = d * b + r.
It immediately follows that
d = c_k * b^{k-1} + ... + c_1 * b^0.
So, to find all digits except the last, we got to solve the same problem but for d < n instead of n.
Overall correctness now follows trivially by induction by n.
