# how-many-zeros-has-the-number-1000-at-the-end-and-what-is-the-last-digit-before-these-zeros-

Question Number 134034 by mr W last updated on 27/Feb/21
$${how}\:{many}\:{zeros}\:{has}\:{the}\:{number} \\$$$$\mathrm{1000}!\:{at}\:{the}\:{end}?\:{and}\:{what}\:{is}\:{the} \\$$$${last}\:{digit}\:{before}\:{these}\:{zeros}? \\$$
Answered by floor(10²Eta[1]) last updated on 27/Feb/21
$$\lfloor\frac{\mathrm{1000}}{\mathrm{5}}\rfloor+\lfloor\frac{\mathrm{1000}}{\mathrm{5}^{\mathrm{2}} }\rfloor+\lfloor\frac{\mathrm{1000}}{\mathrm{5}^{\mathrm{3}} }\rfloor+\lfloor\frac{\mathrm{1000}}{\mathrm{5}^{\mathrm{4}} }\rfloor \\$$$$=\mathrm{200}+\mathrm{40}+\mathrm{8}+\mathrm{1} \\$$$$=\mathrm{249}\:\mathrm{zeros} \\$$$$\mathrm{the}\:\mathrm{last}\:\mathrm{digit}\:\mathrm{before}\:\mathrm{the}\:\mathrm{zeros}\:\mathrm{is} \\$$$$\frac{\mathrm{1000}!}{\mathrm{10}^{\mathrm{249}} }\left(\mathrm{mod10}\right)=? \\$$
Commented by mr W last updated on 27/Feb/21
$$\mathrm{100}\:{has}\:{two}\:{times}\:\mathrm{10}.\:{you}\:{counted}\:{only}\:{one}\:{time}. \\$$$$\mathrm{1000}\:{has}\:{three}\:{times}\:\mathrm{10}.\:{you}\:{counted}\:{only}\:{one}\:{time}. \\$$$$\mathrm{50},\:\mathrm{150},\:\mathrm{250}…\:{each}\:{forms}\:{a}\:\mathrm{10}\:{and}\:{a}\:\mathrm{5}.\:{you}\:{counted}\:{only}\:\mathrm{10}. \\$$
Commented by mr W last updated on 27/Feb/21
$$\mathrm{249}\:{zeros}\:{are}\:{correct}. \\$$
Commented by malwan last updated on 27/Feb/21
$${what}\:{is}\:{missing}\:{in}\:{my}\:{method}\:{sir}? \\$$
Commented by malwan last updated on 28/Feb/21
$${yes}\:{sir} \\$$$${thank}\:{you}\:{so}\:{much} \\$$