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Question Number 20907 by DKumar last updated on 07/Sep/17
How many zeroes (0) there in 1×2×3×4..........99×100
$${How}\:{many}\:{zeroes}\:\left(\mathrm{0}\right)\:{there}\:{in}\:\mathrm{1}×\mathrm{2}×\mathrm{3}×\mathrm{4}……….\mathrm{99}×\mathrm{100}\: \\ $$
Answered by dioph last updated on 07/Sep/17
zeros in number = maximum integer  n such that 10^n =2^n 5^n  divides the number.  As there is plenty more factors of 2 than  factors of 5, we will find n = number  of factors of 5.  There is one for each multiple of 5  between 1 and 100 (20 of them) +  another one for each multiple of  25 between 1 and 100 (4 of them).  Hence, 100! ends in 24 zeros.
$$\mathrm{zeros}\:\mathrm{in}\:\mathrm{number}\:=\:\mathrm{maximum}\:\mathrm{integer} \\ $$$${n}\:\mathrm{such}\:\mathrm{that}\:\mathrm{10}^{{n}} =\mathrm{2}^{{n}} \mathrm{5}^{{n}} \:\mathrm{divides}\:\mathrm{the}\:\mathrm{number}. \\ $$$$\mathrm{As}\:\mathrm{there}\:\mathrm{is}\:\mathrm{plenty}\:\mathrm{more}\:\mathrm{factors}\:\mathrm{of}\:\mathrm{2}\:\mathrm{than} \\ $$$$\mathrm{factors}\:\mathrm{of}\:\mathrm{5},\:\mathrm{we}\:\mathrm{will}\:\mathrm{find}\:{n}\:=\:\mathrm{number} \\ $$$$\mathrm{of}\:\mathrm{factors}\:\mathrm{of}\:\mathrm{5}. \\ $$$$\mathrm{There}\:\mathrm{is}\:\mathrm{one}\:\mathrm{for}\:\mathrm{each}\:\mathrm{multiple}\:\mathrm{of}\:\mathrm{5} \\ $$$$\mathrm{between}\:\mathrm{1}\:\mathrm{and}\:\mathrm{100}\:\left(\mathrm{20}\:\mathrm{of}\:\mathrm{them}\right)\:+ \\ $$$$\mathrm{another}\:\mathrm{one}\:\mathrm{for}\:\mathrm{each}\:\mathrm{multiple}\:\mathrm{of} \\ $$$$\mathrm{25}\:\mathrm{between}\:\mathrm{1}\:\mathrm{and}\:\mathrm{100}\:\left(\mathrm{4}\:\mathrm{of}\:\mathrm{them}\right). \\ $$$$\mathrm{Hence},\:\mathrm{100}!\:\mathrm{ends}\:\mathrm{in}\:\mathrm{24}\:\mathrm{zeros}. \\ $$

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