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Lesson 8 of 8

Keep the arithmetic contract explicit

Connect totients, huge exponents, congruences, and division-free sums.

Understand the idea

Euler’s totient φ(n) counts values in [1,n] coprime to n and equals n times the product of (1 − 1/p) over distinct prime factors. To process a decimal exponent too large to store, update result = result^10 × base^digit modulo m for each digit.

Trace a small example

φ(12) = 4, corresponding to 1, 5, 7, 11. A geometric sum can be combined alongside a power using doubling identities, without assuming a − 1 has an inverse. Coprime congruences modulo 3 and 5 combine into one residue modulo 15.

Conditions and cost

Euler exponent reduction requires coprimality; decimal digit processing does not. Simple Chinese-remainder formulas require coprime moduli. For larger modulus bounds, ordinary long long multiplication may need an overflow-safe replacement.

Pause & predict

When is a^φ(m) congruent to 1 modulo m valid for m > 1?

Choose an answer
Try it yourself

Turn the idea into code

Find the residue modulo 15 that is 2 modulo 3 and 3 modulo 5.

Give me a hint

Test 3, 8, 13 against modulus 3.

Show the explained solution

8 satisfies both conditions. Coprimality makes the solution unique modulo 15.

Your notes

Ready for the next idea?

Mark complete when you can explain it yourself.

From a problem to a programIdentify the input, output, and constraints, then translate a small plan into C++.Variables, types, and remembered valuesChoose a representation, initialize it, and understand what assignment really changes.Arithmetic without surprisesPredict division and remainder, widen before multiplication, and format decimal answers.Conditions and logical decisionsBuild exclusive categories, combine comparisons, and protect operations with short-circuit logic.Optional: other ways to chooseRecognize switch for exact choices and the conditional operator for choosing a value.For loops: make repetition preciseTrace initialization, condition, body, and update—and avoid an extra iteration.While, do-while, and stopping rulesChoose a loop from its stopping condition and read the exact input format.Break, continue, and early answersDistinguish leaving a loop from skipping one iteration, without losing required input.Five patterns worth understandingUse sums, counts, best-so-far candidates, products, and flags to process a stream.Nested loops: rows and columnsTranslate a shape into two loop roles and count the total work.Take a number apart, one digit at a timeUse remainder and division to process decimal digits, including zero.Multiple cases and a debugging mindsetCombine the patterns, isolate each case, and test the boundaries that can break your idea.Arrays: give every value a placeUnderstand indices, initialization, capacity, active length, and the first safe array loop.Scan, count, search, and compareDistinguish values from positions and preserve the exact tie rule a problem asks for.Change a sequence without losing itUpdate, copy, compare, reverse, and inspect neighbours using the correct bounds.Matrices: think in two coordinatesTraverse tables, total rows and columns, inspect diagonals, and protect grid neighbours.Characters become stringsSeparate characters from numbers, then safely inspect and edit managed text.Read the text the problem actually givesChoose words or full lines, preserve empty cases, and reset only the state that needs it.Scan text: count, filter, transformBuild a result from matching characters without losing leading zeros or skipping shifted text.Join, compare, and find textLearn content comparison, lexicographical order, substring counts, and the not-found sentinel.