Trailing Zeros of 100!

Question

How many zeros does 100! end in? Compute it mentally, and be ready to defend the method, because the obvious first answer is wrong.

Accepted Answer

Each trailing zero needs a factor 10 = 2 times 5. Factors of 2 are plentiful, so the count of 5s is the bottleneck. Count powers of 5 in 100! by Legendre's formula: leftlfloor frac1005 rightrfloor + leftlfloor frac10025 rightrfloor + leftlfloor frac100125 rightrfloor = 20 + 4 + 0 = mathbf24. The lfloor 100/25 rfloor term exists because 25, 50, 75, 100 each contribute a second five, forgetting it (answering 20) is the classic miss. Why a trading firm asks this: it tests whether you decompose a fuzzy question ("how many zeros?") into the exact scarce resource (factors of 5), the same instinct as finding the binding constraint in a trade.

Trailing Zeros of 100!

Problem

Your answer

Hints

More Combinatorics to train on