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Lesson 3 of 8

Keep residues and normalize them

Preserve remainders under addition and multiplication, and treat division separately.

Understand the idea

For m > 0, congruent integers share a remainder class. Addition, subtraction, and multiplication preserve congruence. In C++, normalize a negative remainder using x %= m, then add m if x < 0.

Trace a small example

−3 modulo 7 has canonical residue 4. A huge decimal integer can be processed digit by digit using r = (10r + digit) mod m. For 12345 modulo 7 the final remainder is 4. Modular prefix queries subtract and normalize.

Conditions and cost

Bound intermediate sums and products before evaluating them: taking % after overflow is too late. Ordinary division of residues does not represent modular division. That operation requires an inverse.

Pause & predict

What canonical residue represents −3 modulo 7?

Choose an answer
Try it yourself

Turn the idea into code

Compute the canonical residue of 4 − 9 modulo 6.

Give me a hint

Normalize the remainder of −5.

Show the explained solution

The answer is 1 because −5 + 6 = 1.

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.