Zeros at the end of 100!
How many zeros does the decimal expansion of end with?
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
Each trailing zero is a factor . Which of the two primes is the bottleneck?
Count the factors of 5 in . Some numbers contribute more than one.
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s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. The number of trailing zeros is the number of times 10 divides . There are far more factors of 2 than of 5, so count the factors of 5.
Step 1. Multiples of 5 up to 100: of them, each giving one factor 5.
Step 2. Multiples of 25 (25, 50, 75, 100) give a second factor 5: more.
Step 3. No multiple of 125 is at most 100, so the total is
The trap. Answering 20 by forgetting that 25, 50, 75 and 100 each contain . For the count is .