COSMO · STUDY + TEACHING DECK

Number Theory

November 5, 2025

A question worth sharing

Fermat and Euler Theorems

Use multiplicative cycles to reduce enormous exponents modulo n.

ThinkCompareExplain

Today’s route

A question becomes an explanation.

Try

Enter through a problem. A partial attempt counts.

Connect

Use totient, unit, order 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

Find $3^{100}$ mod 7.

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.

totient

count of residues coprime to n

unit

residue with a multiplicative inverse

order

first positive exponent returning 1

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

The core idea

Keep the idea. Explain the conditions.

\gcd(a,n)=1\quad\Longrightarrow\quad a^{\varphi(n)}\equiv1\pmod n

The idea

Use multiplicative cycles to reduce enormous exponents modulo n.

What to remember

A huge exponent becomes a position inside a finite multiplicative cycle.

What information does this method keep?

Worked example

One route, one decision at a time

The problem

Compute $7^{222}$ mod 10.

Predict each next step

The last-digit cycle is 7,9,3,1.
Its length is 4.
222 = 2 mod 4.
Use the second entry: 9.

Which step needs a reason, not just a calculation?

See the structure

The structure behind fermat and euler theorems

\gcd(a,n)=1\quad\Longrightarrow\quad a^{\varphi(n)}\equiv1\pmod n

The coprimality condition permits exponent reduction; always check it first.

Read the representation

Use multiplicative cycles to reduce enormous exponents modulo n.

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 fermat and euler theorems

Your shared task

Students build power cycles for several bases and compare the observed order with phi(n).

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 huge exponent becomes a position inside a finite multiplicative cycle.
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

Compute $2^{100}$ mod 15.

Predict each next step

phi(15) = 8 and gcd(2,15) = 1.
So $2^{8}$ = 1 mod 15.
100 = 4 mod 8.
$2^{4}$ = 16 = 1 mod 15.

Which step needs a reason, not just a calculation?

Convince a skeptic

Turn a pattern into a proof.

The claim

If gcd(a,n) = 1, then $a^{phi}$ (n) = 1 mod n.

Build the argument

Multiplication by a permutes the reduced residue classes.
The product of the permuted classes equals the original product.
Cancel the invertible residue product to obtain the theorem.

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

Stretch together

Take the idea somewhere new.

Transfer challenge

Find the last two digits of $7^{2026}$ .

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

Exponent reduction supports cryptography, primality tests, and fast modular computation.

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

Find $3^{100}$ mod 7.

Question 2

Explain why this claim is true: If gcd(a,n) = 1, then $a^{phi}$ (n) = 1 mod n.

Question 3

Find the last two digits of $7^{2026}$ .

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 huge exponent becomes a position inside a finite multiplicative cycle.

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

A good question belongs to everyone in the room.