| -----------------------------------------------------------------------
Algorithm E (Euclid's algorithm). Given two positive integers m and n,
find their greatest common divisor, that is, the largest positive integer
that evenly divides both m and n. E1 [Find remained] Divide m by n and r be the remainder. (We will
know that 0 <= r < n) E2 [Is it zero?] If r = 0, the algorithm terminates, n is the answer E3 [Reduce] Set m = n, then n = r, then go back to step E1
----------------------------------------------------------------------- [3] Change Algorithm E so that all trivial replacement operations
such as m = n are avoided. Write this new algorithm in the style of
Algorithm E and call it Algorithm F. ========================================================================
Claude Sonnet 4.6
======================================================================== Algorithm F Given two positive integers m and n, find their greatest common divisor. F1 [Find remainder] Divide m by n and let r be the remainder.
(We will know that 0 ≤ r < n.) F2 [Is it zero?] If r = 0, the algorithm terminates and n is the answer. F3 [Find remainder] Divide n by r and let m be the remainder.
(We will know that 0 ≤ m < r.) F4 [Is it zero?] If m = 0, the algorithm terminates and r is the answer. F5 [Find remainder] Divide r by m and let n be the remainder.
(We will know that 0 ≤ n < m.) Go back to F2. Explanation The key insight is that instead of shuffling values with temporary
replacements like: m = n, n = r we reuse the existing variables (m, n, r) in rotation, dividing and
storing each new remainder directly into the variable that is no
longer needed: Step Divides Remainder stored in
F1...m by n...r
F3...n by r...m
F5...r by m...n Each variable naturally becomes the "oldest" value at each stage and
is safely overwritten. This eliminates all trivial assignments while
preserving the logic of Euclid's algorithm. □ |