COSMO · STUDY + TEACHING DECK

Number Theory

October 9, 2025

A question worth sharing

Linear Diophantine Equations

Decide when ax + by = c has integer solutions and describe every solution.

ThinkCompareExplain

Today’s route

A question becomes an explanation.

Try

Enter through a problem. A partial attempt counts.

Connect

Use Diophantine equation, solvability, parameter to name what you notice.

Convince

Build an argument a partner can check.

The goal is a method someone else can use.

The opener

Start with a question.

Opening problem

Can 18x + 30y = 7 have integer solutions?

Think · 2 min

Sketch, test a small case, or make a first guess.

Pair · 3 min

Compare methods. Find one thing you agree on and one question.

Be ready to share your route, even if it is unfinished.

Compare before checking

Put two attempts in conversation.

Explain

Tell a partner how you started. Let them finish before responding.

Question

Which step feels certain? Which step still needs evidence?

Build

Combine one useful idea from each attempt into a new route.

Can you explain your partner’s method in your own words?

Board conversation

Notice before naming.

Look for structure

Try a small case and record every step.
Separate what changes from what stays fixed.
State the pattern in a sentence before naming the theorem.

Make a conjecture

We think ___ because ___. A case that might break this is ___.

What changed? What stayed the same?

Useful language

Now the language has a job.

Diophantine equation

equation seeking integer solutions

solvability

whether a solution exists

parameter

integer controlling a family

Use one of these words to explain something you already noticed.

The core idea

Keep the idea. Explain the conditions.

ax+by=c\text{ has }x,y\in\mathbb Z\quad\Longleftrightarrow\quad\gcd(a,b)\mid c

The idea

Decide when ax + by = c has integer solutions and describe every solution.

What to remember

A gcd condition decides existence; one parameter describes the whole solution family.

What information does this method keep?

Worked example

One route, one decision at a time

The problem

Solve 14x + 21y = 35 in integers.

Predict each next step

Divide by 7 to get 2x + 3y = 5.
One solution is x = 1, y = 1.
Adding 3t to x and subtracting 2t from y preserves the equation.
All solutions are x = 1 + 3t, y = 1 - 2t, for integer t.

Which step needs a reason, not just a calculation?

See the structure

The structure behind linear diophantine equations

ax+by=c\text{ has }x,y\in\mathbb Z\quad\Longleftrightarrow\quad\gcd(a,b)\mid c

For integer a and b not both zero, the gcd decides whether integer solutions exist.

Read the representation

Decide when ax + by = c has integer solutions and describe every solution.

Look together

Point to the part that explains the relationship. Change one input and predict what changes.

What can you see here that was harder to see in words?

Two approaches

Compare the reasoning.

Structural method

Name the objects and conditions.
Use a reusable relationship.
Explain why the method must work.

Brute force

May help with a small test case.
Can hide the general reason.
Becomes unreliable when the scale grows.

Which route can you justify? What would you test first?

Reasoning clinic

Find the step that needs repair.

Investigate these concerns

May help with a small test case.
Can hide the general reason.
Becomes unreliable when the scale grows.

Repair together

Name the missing condition. Try a small counterexample. Rewrite one step so it is justified.

A useful correction explains why the original step fails.

Work together

Investigate linear diophantine equations

Your shared task

Groups model stamp, ticket, or coin totals and use modular arithmetic to restrict the integer solutions.

Explore

4 min - test cases

Compare

5 min - form a rule

Defend

4 min - defend the rule

Everyone should be able to explain the group’s final claim.

Share → connect

What did the room discover?

Bring your evidence

Share a conjecture, a useful example, or a route that changed.

Connections to investigate

A gcd condition decides existence; one parameter describes the whole solution family.
A complete solution names the condition that makes the method valid.
A counterexample is useful when it exposes a missing condition.

Which observation explains the mechanism?

Try → explain → check

Your turn to build the route

The problem

Find all nonnegative integer solutions to 4x + 7y = 39.

Predict each next step

Reduce mod 4: 3y = 3 mod 4.
Thus y = 1 mod 4 and nonnegativity gives 0 ≤ y ≤ 5.
The possible values are y = 1 and y = 5.
They give (x,y) = (8,1) and (1,5).

Which step needs a reason, not just a calculation?

Convince a skeptic

Turn a pattern into a proof.

The claim

For integers a,b not both zero, ax + by = c has integer solutions exactly when gcd(a,b) divides c.

Build the argument

Every linear combination ax + by is divisible by gcd(a,b).
Bezout gives gcd(a,b) as one linear combination.
Multiply that identity by c/gcd(a,b) when the divisibility condition holds.

What does each step rely on? Where are the conditions used?

Stretch together

Take the idea somewhere new.

Transfer challenge

A theater sells $7 and$ 11 tickets and collects $145. Find all nonnegative ticket counts.

Choose a route

Name the objects, the conditions, and a relationship that might help. Start with one test case.

What transfers from the earlier example? What is different?

Peer review

Check the argument together.

Restate

Explain the other person’s route before evaluating it.

Test

Try a small case or boundary case. Does every step still make sense?

Revise

Fix an assumption, add a reason, or find a second method.

Where did your thinking change?

Beyond this problem

Where might this idea help?

A connection

Integer equations model packaging, schedules, exact totals, and resource combinations.

Discuss the model

What objects or quantities are represented?
Which relationship from today is preserved?
What assumption would you check before trusting it?

Invent one more situation with the same structure.

Exit ticket

Leave with an explanation.

Question 1

Can 18x + 30y = 7 have integer solutions?

Question 2

Explain why this claim is true: For integers a,b not both zero, ax + by = c has integer solutions exactly when gcd(a,b) divides c.

Question 3

A theater sells $7 and$ 11 tickets and collects $145. Find all nonnegative ticket counts.

Choose one. Write an answer with a reason, then compare with a partner.

Reflect together

Make your thinking visible.

I used to think…

Name a claim or approach that changed.

Now I can explain…

Give the reason, not only the result.

I still wonder…

Write a question worth bringing to the next circle.

Share one sentence. Ask one follow-up question.

The idea to keep

A gcd condition decides existence; one parameter describes the whole solution family.

Use the matching practice sets to move from direct examples to proof and transfer.

A good question belongs to everyone in the room.