Diophantine Equations

• What a linear Diophantine equation ax + by = c is, and exactly when it has integer solutions.
• How to generate all integer solutions once you have found one, using the pattern x = x₀ + (b/g)t, y = y₀ − (a/g)t.
• How to solve classic olympiad "coin problem" and "make exact change" puzzles using this machinery.

1. Solvability test — ax + by = c has integer solutions if and only if gcd(a, b) divides c. If gcd(a, b) = g and g ∤ c, there are no integer solutions at all.
2. One particular solution — once you spot or compute (via the Euclidean algorithm) one solution (x₀, y₀), every other integer solution is x = x₀ + (b/g)t, y = y₀ − (a/g)t for any integer t.
3. Non-negative solutions — many olympiad problems ask only for solutions with x ≥ 0 and y ≥ 0 (e.g. "coins" or "objects" can't be negative), which restricts t to a finite range.
4. Frobenius idea (two coprime coin values) — if a and b are coprime positive integers, the largest amount that cannot be made using only non-negative multiples of a and b is ab − a − b.

Find all non-negative integer solutions of 3x + 5y = 19.

Step 1 — gcd(3,5) = 1, and 1 divides 19, so solutions exist
Step 2 — spot one solution by trying small y: y=2 → 3x=9 → x=3. So (x0,y0)=(3,2)
Step 3 — general solution: x = 3 + 5t, y = 2 - 3t
Step 4 — need x ≥ 0 and y ≥ 0: t=0 gives (3,2); t=-1 gives x=-2 (invalid);
          y ≥ 0 needs t ≤ 0 (since 2-3t≥0 → t≤2/3); x≥0 needs t≥-3/5
Step 5 — only integer t in [-3/5, 2/3] is t=0
Answer — the only non-negative solution is x=3, y=2

• Trying to solve ax + by = c when gcd(a, b) does not divide c — no solution exists, so don't hunt endlessly.
• Forgetting the ± sign pattern: x increases by b/g while y decreases by a/g for each step of t (not both increasing).
• Assuming a single particular solution is the only solution — always write the general family before restricting to non-negative values.

• Does 6x + 9y = 20 have integer solutions? Why not? (gcd(6,9)=3, and 3 does not divide 20.)
• Find one non-negative integer solution of 4x + 7y = 15 (x=2, y=1 works: 8+7=15).
• Using coins of value 3 and 5 (coprime), what is the largest amount you can never make exactly? (3×5 − 3 − 5 = 7.)

Open the Practice tab for graded questions on Diophantine Equations.

• Coin-combination simulator: pick two coin/note denominations and let students search for which totals are makeable, building intuition for the Frobenius number.
• Mirror / body / home activity: using real coins of two denominations at home (e.g. ₹2 and ₹5), list every total from ₹1 to ₹20 that can and cannot be made exactly.
• Voice or text reflection with AI Mentor: explain why a shopkeeper giving "exact change only" with two coin types sometimes cannot fulfill an order.

• "Explain the coin problem to a Class 6 student using ₹2 and ₹5 coins from an Indian shop."
• "What is one common mistake students make when generating all solutions of a linear Diophantine equation, and how would you catch yourself making it?"
• Stretch: "How does this connect to giving exact change, packaging in fixed box sizes, or scheduling problems?"

• Complete the core practice + one extension activity (photo, table, short reflection, or mini-project) for base XP + topic badge.
• 5-7 day streak or family discussion note = multiplier + visible artifact in parent/principal dashboard.
• Best real-world application stories (anonymised) featured on class or national leaderboard.

• One hands-on project or measurement using the Drishti kit or household items that makes the concept physical.
• Direct link to at least one Future Skill track (Money Management, Green Tech, Cyber Defenders, Micro-Entrepreneurship, AI Mastery, Sustainable Living, Personality Development).
• Coding extension where relevant (simple script, simulation, or data logging).

This material emphasises experiential "learning by doing", competency (apply/create/analyse), vocational exposure, critical thinking, and multidisciplinary connections. Designed to feed live worlds, AI Mentor (with memory), gamification, robotics, parent analytics, and future skills — not just exam prep.

Portfolio Evidence Idea: Your photo/table/reflection/project + one sentence on "How this helps me in real life or a possible future path."

Open the Practice tab for aligned questions (easy/medium/hard + case-based) with full AI scaffolding.

See curriculum for cross-links and the full future-skills/robotics chapters.