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

⌊((1000)/5)⌋+⌊((1000)/5^2 )⌋+⌊((1000)/5^3 )⌋+⌊((1000)/5^4 )⌋ =200+40+8+1 =249 zeros the last digit before the zeros is ((1000!)/(10^(249) ))(mod10)=?

$$\lfloor\frac{\mathrm{1000}}{\mathrm{5}}\rfloor+\lfloor\frac{\mathrm{1000}}{\mathrm{5}^{\mathrm{2}} }\rfloor+\lfloor\frac{\mathrm{1000}}{\mathrm{5}^{\mathrm{3}} }\rfloor+\lfloor\frac{\mathrm{1000}}{\mathrm{5}^{\mathrm{4}} }\rfloor \\ $$$$=\mathrm{200}+\mathrm{40}+\mathrm{8}+\mathrm{1} \\ $$$$=\mathrm{249}\:\mathrm{zeros} \\ $$$$\mathrm{the}\:\mathrm{last}\:\mathrm{digit}\:\mathrm{before}\:\mathrm{the}\:\mathrm{zeros}\:\mathrm{is} \\ $$$$\frac{\mathrm{1000}!}{\mathrm{10}^{\mathrm{249}} }\left(\mathrm{mod10}\right)=? \\ $$

Commented by mr W last updated on 27/Feb/21

100 has two times 10. you counted only one time. 1000 has three times 10. you counted only one time. 50, 150, 250... each forms a 10 and a 5. you counted only 10.

$$\mathrm{100}\:{has}\:{two}\:{times}\:\mathrm{10}.\:{you}\:{counted}\:{only}\:{one}\:{time}. \\ $$$$\mathrm{1000}\:{has}\:{three}\:{times}\:\mathrm{10}.\:{you}\:{counted}\:{only}\:{one}\:{time}. \\ $$$$\mathrm{50},\:\mathrm{150},\:\mathrm{250}…\:{each}\:{forms}\:{a}\:\mathrm{10}\:{and}\:{a}\:\mathrm{5}.\:{you}\:{counted}\:{only}\:\mathrm{10}. \\ $$

Commented by mr W last updated on 27/Feb/21