Proof portfolio: from a remainder procedure to a theorem: changed case
Address this changed case in a fresh argument. Explain which earlier conclusions survive and which require new evidence. This is public practice with available guidance; it is not an untouched examination. Your responses and guidance exposure are stored separately from the main project.
Loading saved investigation…
0 of 1 transfer tasks have recorded responses. This counts submissions, not quality, independent review or project mastery.
Changed case: solve and classify a congruence
For an integer a, positive integer m and integer c, define ax≡c (mod m) to mean m divides ax−c. Starting from your portfolio theorem, prove that a solution exists exactly when d=gcd(a,m) divides c. If solutions exist, prove there are exactly d distinct solution classes modulo m, and describe all integer solutions. Apply this to 18x≡12 (mod 30), then explain why 18x≡5 (mod 30) fails. Your proof must establish both completeness and distinctness of the classes, including a=0 and m=1. Record prior exposure to this public guidance; for independent transfer a reviewer should supply a different formulation.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Review, defend and revise
Ask a reviewer to examine the complete reasoning, reproduce selected results and pose an unfamiliar case. Record their feedback against the specific revision discussed. Reviewer labels are entered locally and do not verify identity or judgment. A complete set of text submissions does not certify an independently defended investigation.
