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

....number theory... Question : If a , b , c ∈ N ; then prove ::: a!∗b!∗c!∣(a+b+c)! m.n . july 970#

$$\:\:\:\:\:....{number}\:{theory}... \\ $$$$\:\:\:\:\:\:\:{Question}\::\:\:\:\:\:\:\mathrm{I}{f}\:\:\:{a}\:,\:{b}\:,\:{c}\:\:\in\:\mathbb{N}\:\:\:;\:{then}\: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:{prove}\:::: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:{a}!\ast{b}!\ast{c}!\mid\left({a}+{b}+{c}\right)!\:\: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{m}.{n}\:.\:{july}\:\mathrm{970}# \\ $$

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

Oh! Sorry, I thought it was an equation.

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

Commented by MJS_new last updated on 08/Sep/20

then please give a counterexample

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

Answered by MJS_new last updated on 08/Sep/20

just a try... let a≤b≤c; obviously c!∣(a+b+c)! (((a+b+c)!)/(c!))=(c+1)(c+2)...(c+a+b) the number of the factors = a+b b!=1×2×3×...×b now each factor of b! is included at least once in b factors of the form (c+1)(c+2)...(c+b) this should be easy to see: if c=b+n ⇒ b∣(c+b−n); 1≤n≤b the remaining question is, how to show the “remaining” number is divisible by a let b+c=n then (((a+n)!)/(n!))=(n+1)(n+2)...(a+n) same argument as above

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

Commented by mnjuly1970 last updated on 09/Sep/20

thank you sir.excellent and admirable..

$${thank}\:{you}\:{sir}.{excellent} \\ $$$${and}\:{admirable}.. \\ $$

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