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Question Number 115294 by ZiYangLee last updated on 24/Sep/20

$$\mathrm{Given}\mathrm{that}N=1\times 2\times 3\times ...\times 500\mathrm{is}\mathrm{the}$$$$\mathrm{product}\mathrm{of}\mathrm{the}\mathrm{positive}\mathrm{integers}\mathrm{from}1\mathrm{to}500.$$$$$$$$\mathrm{If}N\mathrm{is}\mathrm{divisible}\mathrm{by}{6}^k,\mathrm{find}\mathrm{the}\mathrm{largest}\mathrm{possible}$$$$\mathrm{value}\mathrm{of}k.$$

Answered by Olaf last updated on 25/Sep/20

$$\mathrm{N}=500!$$$$2,4,6,8...498,500\mathrm{are}\mathrm{divisible}\mathrm{by}2$$$$\left(250\mathrm{numbers}\right)$$$$3,6,9,12...495,498\mathrm{are}\mathrm{divisible}\mathrm{by}3$$$$3=3\times 1$$$$6=3\times 2$$$$...$$$$498=3\times 166$$$$\left(166\mathrm{numbers}\right)$$$$$$$$250\mathrm{numbers}\mathrm{are}\mathrm{divisible}\mathrm{by}2$$$$166\mathrm{numbers}\mathrm{are}\mathrm{divisible}\mathrm{by}3$$$$6^k=2^k\times 3^k:k_{max}=166$$$$\mathbf{Sorry}\mathbf{but}{\mathbf{i}}^{\prime }\mathbf{m}\mathbf{wrong}.$$$$\mathbf{see}\mathbf{mr}\mathbf{W}.$$

Answered by mr W last updated on 25/Sep/20

$$2,2\times 2,3\times 2,...,250\times 2\Rightarrow 2^{250\times 1}$$$$2^2,2\times 2^2,3\times 2^2,...,125\times 2^2\Rightarrow 2^{125\times 2}$$$$2^3,2\times 2^3,3\times 2^3,...,62\times 2^3\Rightarrow 2^{62\times 3}$$$$2^4,2\times 2^4,3\times 2^4,...,31\times 2^4\Rightarrow 2^{31\times 4}$$$$2^5,2\times 2^5,3\times 2^5,...,15\times 2^5\Rightarrow 2^{15\times 5}$$$$2^6,2\times 2^6,3\times 2^6,...,7\times 2^6\Rightarrow 2^{7\times 6}$$$$2^7,2\times 2^7,3\times 2^7,...,3\times 2^7\Rightarrow 2^{3\times 7}$$$$2^8\Rightarrow 2^{1\times 8}$$$$250+125+62+31+15+7+3+1=494$$$$\Rightarrow 500!contains{2}^{494}$$$$$$$$3,2\times 3,3\times 3,...,166\times 3\Rightarrow 3^{166\times 1}$$$$3^2,2\times 3^2,3\times 3^2,...,55\times 3^2\Rightarrow 3^{55\times 2}$$$$3^3,2\times 3^3,3\times 3^3,...,18\times 3^3\Rightarrow 3^{18\times 3}$$$$3^4,2\times 3^4,3\times 3^4,...,6\times 3^4\Rightarrow 3^{6\times 4}$$$$3^5,2\times 3^5\Rightarrow 3^{2\times 5}$$$$166+55+18+6+2=247$$$$\Rightarrow 500!contains{3}^{247}$$$$$$$$\Rightarrow 500!contains{2}^{494}\times 3^{247}$$$$\Rightarrow 500!contains{6}^{247}$$$$\Rightarrow k_{max}=247$$

Commented by Olaf last updated on 25/Sep/20

$$\mathrm{Mister}\mathrm{W}\mathrm{you}\mathrm{are}\mathrm{wright}.$$$$\mathrm{Good}\mathrm{job}!$$

Commented by mr W last updated on 25/Sep/20

$$ithinkwecangenerallydolikethis:$$$$togetksuchthatn!isdivisibleby$$$$p^{k}wherep=prime,$$$$k=\lfloor \frac{n}{p}\rfloor +\lfloor \frac{n}{p^2}\rfloor +\lfloor \frac{n}{p^3}\rfloor +...$$$$$$$$forexamplen=500,p=7$$$$k=\lfloor \frac{500}{7}\rfloor +\lfloor \frac{500}{7^2}\rfloor +\lfloor \frac{500}{7^3}\rfloor +...$$$$=71+10+1$$$$=82$$$$\Rightarrow 500!contains{7}^{82}$$$$$$$$forexamplen=500,p=3$$$$k=\lfloor \frac{500}{3}\rfloor +\lfloor \frac{500}{3^2}\rfloor +\lfloor \frac{500}{3^3}\rfloor +...$$$$=166+55+18+6+2$$$$=247$$$$\Rightarrow 500!contains{3}^{247}$$

Commented by ZiYangLee last updated on 26/Sep/20

$$\mathrm{Wow}!!!$$

Commented by Lordose last updated on 01/Oct/20

$$\mathrm{Thanks}\mathrm{sir}\mathrm{Mr}\mathrm{W}$$

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