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Question Number 116113 by Lordose last updated on 01/Oct/20

$${\mathbf{what}}^{\prime }\mathbf{s}\mathbf{the}\mathbf{exponent}\mathbf{of}12\mathbf{in}\mathbf{the}$$$$\mathbf{expansion}\mathbf{of}100!$$

Answered by mr W last updated on 01/Oct/20

$$12=2^2\times 3$$$$\lfloor \frac{100}{2}\rfloor +\lfloor \frac{100}{2^2}\rfloor +\lfloor \frac{100}{2^3}\rfloor +...$$$$=50+25+12+6+3+1=97$$$$\lfloor \frac{100}{3}\rfloor +\lfloor \frac{100}{3^2}\rfloor +\lfloor \frac{100}{3^3}\rfloor +...$$$$=33+11+3+1=48$$$$\Rightarrow 100!=2^{97}\times 3^{48}\times ...=2\times {\left(2^2\times 3\right)}^{48}\times ...$$$$={12}^{48}\times ...$$

Commented by mr W last updated on 01/Oct/20

$$seealsoQ115294$$

Commented by bemath last updated on 01/Oct/20

$$\mathrm{sir}\mathrm{help}\mathrm{me}\mathrm{qn}115988$$

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