Patterns in the last digits of powers

AA SLAA HL Solid Also in: Pure maths

Research question

Why do the last digits of 2ⁿ, 3ⁿ, 7ⁿ… repeat in cycles, how long are the cycles of the last one, two and three digits, and can you predict the cycle length without listing them?

Adapt it: change the place, the data or the comparison until the question is yours.

Free: the A–E checklist an examiner uses, by email ↓

Why it makes a good exploration

A question anyone can explore by hand that leads naturally to modular arithmetic and proof. Predicting cycle lengths for the last two digits is a genuine challenge.

The mathematics you'll need

Sequences and periodicity
Modular arithmetic (new: explain)
Proof by induction that a cycle repeats
Order of an element; cycle lengths dividing a common number
HL: Euler's theorem stated, tested and partly proved

Course labels show where a technique sits; using maths from outside your course is fine if you explain it clearly and say it is new to you.

Where the data comes from

No data needed: use a spreadsheet; check against OEIS.

OEIS (On-Line Encyclopedia of Integer Sequences) — Check a sequence you have found and read its known formulas and references.
Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.

Cite every source in a footnote where you use it and in your bibliography. Check the licence of any dataset you download.

A possible outline

Explore last digits of powers for each base.
Explain the cycles with modular arithmetic.
Investigate the last two digits and conjecture cycle lengths.
Prove what you can.
Apply it: find the last digits of a huge power and check it.

Pitfalls that cost marks

Tables without explanation.
Using modular notation before introducing it.
Generalising from too few bases.

Showing personal engagement

Start from a puzzle that interested you (the last digit of 7¹⁰⁰⁰).
Make conjectures for other bases.
Find a base with an unexpectedly short cycle.

See Criterion C: personal engagement for what examiners look for.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Good fit	Sequences and periodicity; Modular arithmetic (new: explain)
AA HL	Good fit	Sequences and periodicity; Modular arithmetic (new: explain)
AI SL	Not a natural fit	The mathematics is mainly AA or HL (calculus or proof beyond AI SL); an AI SL version would need a data-driven, technology-based approach.
AI HL	Not a natural fit	The mathematics is mainly from the AA course; an AI HL version would need modelling with technology, statistics or networks at HL level.

Level: Solid. Needs some independent work beyond class examples. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Personal engagement (C)

Build and break your own small cipher or code, invent examples to test each result, and record the conjectures you made and the ones that turned out to be false.

Reflection (D)

Reflect on what each result guarantees and what it doesn't: which errors a check digit misses, which attacks a cipher survives, and how the answer depends on the size of the numbers. For this idea, start with: tables without explanation — say how it affects your answer.

Use of mathematics (E)

SL: Counting principles, probability or frequency statistics used correctly; any number theory (modular arithmetic, primes) introduced with your own small worked examples and explained, not quoted.

HL: Rigorous proofs (by contradiction or induction) of the number-theory facts you rely on, counting arguments made general, or a statistical attack tested formally.

Criteria A and B (presentation and communication) work the same way for every idea: see the guides to Criterion A and Criterion B.

Taking it further

Investigate last digits in other number bases, or prove Fermat's little theorem.

Extending it for HL

This idea already has HL mathematics in it: Euler's theorem stated, tested and partly proved. Prove the key result in general (why the check digit catches every single-digit error, why the decryption undoes the encryption) rather than checking examples.

See a complete IA, marked

Our annotated exemplar How far can a stack of books lean over the edge of a table? (AA HL) asks a different question, but shows how a complete cryptography exploration is structured and marked, with an examiner's comment on every criterion. Free excerpts and the full marking table are on its page.

Before you start: the checklist an examiner uses

Every check for Criteria A–E in a 4-page PDF, the mistakes that cost the most marks and a self-assessment grid. We'll email it with a short IA tip every few days, timed to your deadline if you give it. Free — no account, no payment.

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Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Criteria checklistCheck your plan or draft against the A–E descriptors and self-grade before you hand it in. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, an IB examiner, reads your draft and talks it through with you (€100).