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Lesson 2 of 8

Prove a primality test

Use the smallest-factor argument and respect the input bounds.

Understand the idea

A prime is an integer at least 2 with exactly two positive divisors. Every composite n has a factor at most √n: if both factors exceeded √n, their product would exceed n. Trial division needs only possible divisors through that boundary.

Trace a small example

For 49, testing only d < √n misses 7 and wrongly calls it prime. The inclusive condition d ≤ n/d catches the divisor. Reject 0 and 1 before entering a loop that might do no work.

Conditions and cost

Trial division is O(√n) worst-case. A 64-bit type does not make the loop fast for arbitrary 64-bit inputs. Include the number of queries in the cost estimate. For many bounded queries, prepare a sieve.

Pause & predict

Which value exposes omitting the square-root boundary?

Choose an answer
Try it yourself

Turn the idea into code

Is 97 prime? State the divisors that need testing.

Give me a hint

The integer square-root boundary is 9.

Show the explained solution

Yes. No integer from 2 through 9 divides 97. Testing only primes 2, 3, 5, 7 also suffices if the prime list is complete.

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.