Learning pathOriginal reference
Lesson 7 of 8

Handle several solutions and GCD summaries

Solve linear congruences and combine prefixes with suffixes.

Understand the idea

For ax congruent to b modulo m, set g = gcd(a,m). There is no solution unless g divides b. Otherwise divide a, b, m by g, solve the reduced coprime equation, and recover g residues modulo m separated by m/g.

Trace a small example

6x congruent to 8 modulo 14 reduces to 3x congruent to 4 modulo 7. The reduced answer 6 gives residues 6 and 13 modulo 14. For a different task, GCD after deleting one array element combines the prefix before it and suffix after it.

Conditions and cost

Include O(g) output work when listing all solutions. GCD has no subtraction inverse, so a prefix GCD alone cannot remove an element. Prefix and suffix arrays use O(n) storage and O(n log V) work under a Euclidean bound.

Pause & predict

Does 6x congruent to 5 modulo 14 have a solution?

Choose an answer
Try it yourself

Turn the idea into code

Maximize the GCD of [6,10,15] after deleting one element.

Give me a hint

Check the remaining pair in each case.

Show the explained solution

Deleting 6 yields 5; deleting 10 yields 3; deleting 15 yields 2. The maximum is 5.

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.