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Lesson 5 of 8

Check that an inverse exists

Use coprimality before extended Euclid or the prime-modulus shortcut.

Understand the idea

An inverse x satisfies ax congruent to 1 modulo m. For m ≥ 2 it exists exactly when gcd(a,m) = 1. Extended Euclid finds coefficients ax + my = gcd(a,m); when this is 1, normalize x to obtain the inverse.

Trace a small example

The inverse of 3 modulo 10 is 7 since 21 has residue 1. The value 2 has no inverse modulo 10. For prime p and a not divisible by p, Fermat gives inverse(a) = a^(p − 2) modulo p.

Conditions and cost

The Fermat shortcut does not apply to arbitrary composite moduli or to zero modulo p. Modular a/b means a × inverse(b), with explicit failure if b is not invertible. Bound coefficient-update arithmetic in extended Euclid.

Pause & predict

Does 6 have an inverse modulo 15?

Choose an answer
Try it yourself

Turn the idea into code

Compute 4/3 modulo 7.

Give me a hint

The inverse of 3 modulo 7 is 5.

Show the explained solution

4 × 5 mod 7 = 6. Ordinary integer 4/3 would give an unrelated answer.

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.