Question and Answers Forum

All Questions      Topic List

Arithmetic Questions

Previous in All Question      Next in All Question      

Previous in Arithmetic      Next in Arithmetic      

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.

$$\mathrm{zeros}\:\mathrm{in}\:\mathrm{number}\:=\:\mathrm{maximum}\:\mathrm{integer} \\ $$$${n}\:\mathrm{such}\:\mathrm{that}\:\mathrm{10}^{{n}} =\mathrm{2}^{{n}} \mathrm{5}^{{n}} \:\mathrm{divides}\:\mathrm{the}\:\mathrm{number}. \\ $$$$\mathrm{As}\:\mathrm{there}\:\mathrm{is}\:\mathrm{plenty}\:\mathrm{more}\:\mathrm{factors}\:\mathrm{of}\:\mathrm{2}\:\mathrm{than} \\ $$$$\mathrm{factors}\:\mathrm{of}\:\mathrm{5},\:\mathrm{we}\:\mathrm{will}\:\mathrm{find}\:{n}\:=\:\mathrm{number} \\ $$$$\mathrm{of}\:\mathrm{factors}\:\mathrm{of}\:\mathrm{5}. \\ $$$$\mathrm{There}\:\mathrm{is}\:\mathrm{one}\:\mathrm{for}\:\mathrm{each}\:\mathrm{multiple}\:\mathrm{of}\:\mathrm{5} \\ $$$$\mathrm{between}\:\mathrm{1}\:\mathrm{and}\:\mathrm{100}\:\left(\mathrm{20}\:\mathrm{of}\:\mathrm{them}\right)\:+ \\ $$$$\mathrm{another}\:\mathrm{one}\:\mathrm{for}\:\mathrm{each}\:\mathrm{multiple}\:\mathrm{of} \\ $$$$\mathrm{25}\:\mathrm{between}\:\mathrm{1}\:\mathrm{and}\:\mathrm{100}\:\left(\mathrm{4}\:\mathrm{of}\:\mathrm{them}\right). \\ $$$$\mathrm{Hence},\:\mathrm{100}!\:\mathrm{ends}\:\mathrm{in}\:\mathrm{24}\:\mathrm{zeros}. \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com