Question Number 132386 by liberty last updated on 13/Feb/21 $$\:\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\begin{cases}{\mathrm{ln}\:{i}}\\{\mathrm{ln}\:\left(\mathrm{3}+\mathrm{4}{i}\right)}\end{cases} \\ $$ Answered by EDWIN88 last updated on 13/Feb/21 $$\left(\mathrm{1}\right)\mathrm{ln}\:{i}\:=\:\mathrm{ln}\:\mid{i}\mid\:+{i}\:\left(\mathrm{arg}\:\left({i}\right)+\mathrm{2}{n}\pi\right)=\mathrm{ln}\:\mathrm{1}+{i}\:\left(\frac{\pi}{\mathrm{2}}+\mathrm{2n}\pi\right) \\ $$$$\left(\mathrm{2}\right)\mathrm{ln}\:\left(\mathrm{3}+\mathrm{4}{i}\right)=\mathrm{ln}\:\mid\mathrm{3}+\mathrm{4}{i}\mid=\mathrm{ln}\:\sqrt{\mathrm{3}^{\mathrm{2}} +\mathrm{4}^{\mathrm{2}} }\:+{i}\left(\mathrm{arctan}\:\left(\frac{\mathrm{4}}{\mathrm{3}}\right)+\mathrm{2}{n}\pi\right) \\…
Question Number 1164 by 112358 last updated on 08/Jul/15 $${How}\:{many}\:{five}\:{digit}\:{numbers} \\ $$$${exist}\:{such}\:{that}\:{the}\:{sum}\:{of}\:{their} \\ $$$${digits}\:{equals}\:\mathrm{43}?\: \\ $$$${How}\:{many}\:{exist}\:{if}\:{the}\:{sum}\:{is} \\ $$$$\mathrm{39}? \\ $$$$ \\ $$ Commented by 123456…
Question Number 1156 by 112358 last updated on 06/Jul/15 $${Determine}\:{the}\:{general}\:{solution} \\ $$$${of}\:{the}\:{following}\:\:{linear}\:{diophantine} \\ $$$${equation}\:{for}\:\forall{N}\in\mathbb{Z}^{+} ,{m}\in\mathbb{Z}^{+} : \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{8}{N}=\mathrm{81}{m}+\mathrm{65}\:. \\ $$ Commented by prakash…
Question Number 795 by malwaan last updated on 15/Mar/15 $${what}\:{is}\:{the}\:{last}\:{digit}\:{of} \\ $$$$\mathrm{7}^{\left(\mathrm{7}^{\left(\mathrm{7}….\right)} \right)} \: \\ $$$${the}\:{number}\:{of}\:\mathrm{7}'{s}\:{is}\:\mathrm{1001} \\ $$ Answered by prakash jain last updated on…
Question Number 785 by rishabh last updated on 12/Mar/15 $${Prove}\:{that}\:{if}\:{two}\:{numbers}\:{are}\:{chosen} \\ $$$${at}\:{random}\:{then}\:{the}\:{probability}\:{that} \\ $$$${their}\:{sum}\:{is}\:{divisible}\:{by}\:{n}\:{is}\:\frac{\mathrm{1}}{{n}}. \\ $$ Answered by prakash jain last updated on 12/Mar/15 $$\mathrm{Sum}\:\mathrm{mod}\:{n}={k},\:\mathrm{where}\:\mathrm{0}\leqslant{k}\leqslant{n}−\mathrm{1}.…
Question Number 772 by malwaan last updated on 12/Mar/15 $${prove}\:{that}\: \\ $$$$\mathrm{5555}^{\mathrm{2222}} +\mathrm{2222}^{\mathrm{5555}} \: \\ $$$${is}\:{divisible}\:{by}\:\mathrm{7} \\ $$ Commented by 123456 last updated on 10/Mar/15…
Question Number 769 by rishabh last updated on 09/Mar/15 $${Prove}\:{that}\:{product}\:{of}\:{any}\:\:{n}\:{consecutive}\:{integers} \\ $$$${is}\:{divisiblr}\:{by}\:{n}! \\ $$ Commented by 123456 last updated on 09/Mar/15 $${n}\left({n}+\mathrm{1}\right)\centerdot\centerdot\centerdot\left(\mathrm{2}{n}−\mathrm{1}\right) \\ $$ Answered…
Question Number 765 by prakash jain last updated on 09/Mar/15 $$\mathrm{List}\:\mathrm{all}\:\mathrm{primes}\:{p}\:\mathrm{for}\:\mathrm{which}\:{p}+\mathrm{2}\:\mathrm{and}\:{p}+\mathrm{4} \\ $$$$\mathrm{are}\:\mathrm{also}\:\mathrm{primes}. \\ $$ Answered by rishabh last updated on 09/Mar/15 $${p}=\mathrm{3}\:\mathrm{is}\:\mathrm{the}\:\mathrm{only}\:\mathrm{such}\:\mathrm{number}. \\ $$$$\mathrm{Let}\:\mathrm{us}\:\mathrm{assume}\:\mathrm{there}\:\mathrm{is}\:\mathrm{a}\:\mathrm{prime}\:{p}>\mathrm{3}.…
Question Number 751 by 112358 last updated on 06/Mar/15 $${Determine}\:{the}\:{number}\:{of}\:{integral} \\ $$$${factors}\:{of}\:\mathrm{105840},\:{excluding}\:\mathrm{1}\: \\ $$$${and}\:\mathrm{105840}. \\ $$$$ \\ $$$$ \\ $$ Answered by prakash jain last…
Question Number 66275 by Rasheed.Sindhi last updated on 12/Aug/19 $${Prove}\:{that}\:{for}\:{p},{q},{r}\in\mathbb{N} \\ $$$$\frac{\mathrm{lcm}\left({p},{q},{r}\right)}{\mathrm{gcd}\left({p},{q},{r}\right)}=\frac{{p}×{q}×{r}}{\mathrm{gcd}\left({p},{q}\right)×\mathrm{gcd}\left({q},{r}\right)×\mathrm{gcd}\left({r},{p}\right)} \\ $$ Commented by Prithwish sen last updated on 12/Aug/19 $$\mathrm{p}=\mathrm{nn}_{\mathrm{1}} \mathrm{n}_{\mathrm{2}} \mathrm{x},\mathrm{q}=\mathrm{nn}_{\mathrm{2}}…