Learning pathOriginal reference
Lesson 8 of 8

Extend factorization carefully

Recognize segmented sieves, factorial exponents, and square completion.

Understand the idea

A segmented sieve marks [L,R] using all base primes through √R. For prime p, start at max(p², ceil(L/p) × p). The exponent of p in n! is floor(n/p) + floor(n/p²) + ... .

Trace a small example

The exponent of 2 in 10! is 5 + 2 + 1 = 8. To make n a square with the smallest positive multiplier, include each prime with odd exponent once. For 72 = 2³ × 3², multiply by 2 to obtain 144.

Conditions and cost

Mark 0 and 1 nonprime when a segment includes them. Base primes must cover √R. Repeatedly divide n by p when counting factorial exponents to avoid overflowing powers. The reference also introduces a linear SPF sieve.

Pause & predict

What is the smallest positive m making 45m a square?

Choose an answer
Try it yourself

Turn the idea into code

Find the number of trailing zeros of 100!.

Give me a hint

Count factors of 5; there are more factors of 2.

Show the explained solution

floor(100/5) + floor(100/25) = 20 + 4 = 24, so there are 24 trailing zeros.

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.