Question and Answers Forum

All Questions      Topic List

Set Theory Questions

Previous in All Question      Next in All Question      

Previous in Set Theory      Next in Set Theory      

Question Number 164628 by amin96 last updated on 19/Jan/22

$$60!=\underset{―}{\mathbf{abc}\dots \mathbf{nm}000\dots 0}$$$$\mathbf{m}=?\mathbf{n}=?$$

Commented by amin96 last updated on 20/Jan/22

$$\mathbf{answer}9;6$$$$$$

Answered by Rasheed.Sindhi last updated on 20/Jan/22

$$60!=\underset{―}{\mathbf{abc}\dots \mathbf{nm}000\dots 0}$$$$60!\overline{=abc\dots nm000\dots 0;}m=?n=?$$$${}^{\bullet }\mathrm{Counting}\mathrm{of}\mathrm{factor}5:$$$$\lfloor \frac{60}{5}\rfloor +\lfloor \frac{60}{5^2}\rfloor =12+2=14$$$${}^{\bullet }\mathrm{Counting}\mathrm{of}\mathrm{factor}2:$$$$\lfloor \frac{60}{2}\rfloor +\lfloor \frac{60}{2^2}\rfloor +\lfloor \frac{60}{2^3}\rfloor +\lfloor \frac{60}{2^4}\rfloor +\lfloor \frac{60}{2^5}\rfloor$$$$=30+15+7+3+1=56$$$${}^{\bullet }\frac{60!}{2^{14}\cdot 5^{14}}\overline{=abc\dots nm}$$$$\mathrm{Similar}\mathrm{process}\mathrm{can}\mathrm{help}\mathrm{us}\mathrm{to}\mathrm{determine}$$$$\mathrm{number}\mathrm{of}\mathrm{each}\mathrm{prime}\mathrm{factor}\mathrm{under}$$$$\mathrm{sixty}\mathrm{in}\frac{60!}{2^{14}\cdot 5^{14}}\overline{=abc\dots nm}.{\mathrm{There}}^{\prime }\mathrm{s}$$$$\mathrm{no}\mathrm{factor}5\mathrm{of}\mathrm{course}\mathrm{and}\mathrm{number}\mathrm{of}$$$$2^{\prime }\mathrm{s}\mathrm{is}\mathrm{now}56-14=42$$$$\mathrm{Hence}\frac{60!}{2^{14}\cdot 5^{14}}\overline{=abc\dots nm}\mathrm{now}\mathrm{can}\mathrm{be}$$$$\mathrm{factorized}\mathrm{as}:$$$$2^{42}.3^{28}.7^9.{11}^5.{13}^4.{17}^3.{19}^3.{23}^2$$$$.{29}^2.31.37.41.43.47.53.59$$$$Now,$$$$\frac{60!}{2^{14}\cdot 5^{14}}\overline{=abc\dots nm}\stackrel{100}{\equiv }\overline{nm}$$$${}^{\bullet }{2}^{22}\stackrel{100}{\equiv }4\Rightarrow {\left(2^{22}\right)}^2\stackrel{100}{\equiv }{4}^2=16\Rightarrow 2^{44}\stackrel{100}{\equiv }16$$$$\Rightarrow 2^{44}/2^2\stackrel{100}{\equiv }16/4=4\Rightarrow 2^{42}\stackrel{100}{\equiv }4$$$${}^{\bullet }{3}^{20}\stackrel{100}{\equiv }1\wedge 3^8\stackrel{100}{\equiv }61\Rightarrow 3^{28}\stackrel{100}{\equiv }61$$$${}^{\bullet }{7}^4\stackrel{100}{\equiv }1\Rightarrow {\left(7^4\right)}^2.7\equiv 7\Rightarrow 7^9\stackrel{100}{\equiv }7$$$${}^{\bullet }{11}^5\stackrel{100}{\equiv }51$$$${}^{\bullet }{13}^4\stackrel{100}{\equiv }61$$$${}^{\bullet }{17}^3\stackrel{100}{\equiv }13$$$${}^{\bullet }{19}^3\stackrel{100}{\equiv }59$$$${}^{\bullet }{23}^2\stackrel{100}{\equiv }29$$$${}^{\bullet }{29}^2\stackrel{100}{\equiv }41$$$${}^{\bullet }31\stackrel{100}{\equiv }31$$$${}^{\bullet }37\stackrel{100}{\equiv }37$$$${}^{\bullet }41\stackrel{100}{\equiv }41$$$${}^{\bullet }43\stackrel{100}{\equiv }43$$$${}^{\bullet }47\stackrel{100}{\equiv }47$$$${}^{\bullet }53\stackrel{100}{\equiv }53$$$${}^{\bullet }59\stackrel{100}{\equiv }59$$$$\mathrm{Multiplying}\mathrm{blue}\mathrm{congruences}(\mathrm{for}$$$$\mathrm{convenience}\mathrm{use}\mathrm{modular}\mathrm{multiplication})$$$$\overline{abc\dots nm}\stackrel{100}{\equiv }96$$

Commented by Rasheed.Sindhi last updated on 21/Jan/22

$$\overline{abc\dots nm}\stackrel{100}{\equiv }4.61.7.51.61.13.59.29.41.31.37.41.43.47.53.59$$$$\overline{abc\dots nm}\stackrel{100}{\equiv }(4.61).(7.51).(61.13).(59.29).(41.31).(37.41).(43.47).(53.59)$$$$\stackrel{100}{\equiv }44.57.93.11.71.17.21.27$$$$\stackrel{100}{\equiv }(44.57).(93.11).(71.17).(21.27)$$$$\stackrel{100}{\equiv }08.23.07.67$$$$\stackrel{100}{\equiv }(08.23).(07.67)$$$$\stackrel{100}{\equiv }84.69$$$$\stackrel{100}{\equiv }96$$

Commented by mr W last updated on 21/Jan/22

$$verynicesolution!$$

Commented by Rasheed.Sindhi last updated on 21/Jan/22

$$\mathcal{G}rateful\mathit{s}\mathit{i}\mathit{r}\mathit{m}\mathit{r}\mathit{W}!$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com