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Question Number 112613 by mnjuly1970 last updated on 08/Sep/20

$$....numbertheory...$$$$Question:\mathrm{I}fa,b,c\in \mathbb{N};then$$$$$$$$prove:::$$$$$$$$a!\ast b!\ast c!\mid (a+b+c)!$$$$$$$$\text{You can't use 'macro parameter character \#' in math mode}$$

Commented by Aina Samuel Temidayo last updated on 08/Sep/20

$$\mathrm{Oh}!\mathrm{Sorry},\mathrm{I}\mathrm{thought}\mathrm{it}\mathrm{was}\mathrm{an}$$$$\mathrm{equation}.$$

Commented by MJS_new last updated on 08/Sep/20

$$\mathrm{then}\mathrm{please}\mathrm{give}\mathrm{a}\mathrm{counterexample}$$

Answered by MJS_new last updated on 08/Sep/20

$$\mathrm{just}\mathrm{a}\mathrm{try}...$$$$\mathrm{let}a⩽b⩽c;\mathrm{obviously}c!\mid (a+b+c)!$$$$\frac{(a+b+c)!}{c!}=(c+1)(c+2)...(c+a+b)$$$$\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{the}\mathrm{factors}=a+b$$$$b!=1\times 2\times 3\times ...\times b$$$$\mathrm{now}\mathrm{each}\mathrm{factor}\mathrm{of}b!\mathrm{is}\mathrm{included}\mathrm{at}\mathrm{least}\mathrm{once}$$$$\mathrm{in}b\mathrm{factors}\mathrm{of}\mathrm{the}\mathrm{form}(c+1)(c+2)...(c+b)$$$$\mathrm{this}\mathrm{should}\mathrm{be}\mathrm{easy}\mathrm{to}\mathrm{see}:$$$$\mathrm{if}c=b+n\Rightarrow b\mid (c+b-n);1⩽n⩽b$$$$\mathrm{the}\mathrm{remaining}\mathrm{question}\mathrm{is},\mathrm{how}\mathrm{to}\mathrm{show}\mathrm{the}$$$$‘‘{\mathrm{remaining}}^{″}\mathrm{number}\mathrm{is}\mathrm{divisible}\mathrm{by}a$$$$\mathrm{let}b+c=n$$$$\mathrm{then}\frac{(a+n)!}{n!}=(n+1)(n+2)...(a+n)$$$$\mathrm{same}\mathrm{argument}\mathrm{as}\mathrm{above}$$

Commented by mnjuly1970 last updated on 09/Sep/20

$$thankyousir.excellent$$$$andadmirable..$$

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