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 $$\mathrm{p}=\mathrm{nn}_{\mathrm{1}} \mathrm{n}_{\mathrm{2}} \mathrm{x},\mathrm{q}=\mathrm{nn}_{\mathrm{2}}…
Question Number 591 by 112358 last updated on 04/Feb/15 $${Prove}\:{by}\:{induction}\:{on}\:{n},\:{for}\:{n}\geqslant\mathrm{2}, \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{u}_{{n}} \:\geqslant\:\mathrm{2}^{\mathrm{3}^{{n}−\mathrm{1}} } \\ $$$${for}\:{the}\:{sequence}\:\left\{{u}_{{n}} \right\}\:{defined}\:{by}\: \\ $$$${the}\:{recurrence}\:{relation} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{u}_{\mathrm{1}} =\mathrm{1}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{u}_{{n}+\mathrm{1}} =\left({u}_{{n}}…
Question Number 572 by kth last updated on 29/Jan/15 $${xy}=\mathrm{6}\left({x}+{y}\right) \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} =\mathrm{325} \\ $$$${x}=? \\ $$$${y}=? \\ $$$$ \\ $$ Answered by prakash…
Question Number 558 by 112358 last updated on 26/Jan/15 $${Determine}\:{the}\:{complex}\:{number} \\ $$$${z}\:{such}\:{that}\:{z}^{\mathrm{2}} ={tanx}+{icosx}, \\ $$$${in}\:{the}\:{form}\:{z}={p}+{ik},\:{p},{k}\in{R}. \\ $$$$ \\ $$ Answered by prakash jain last updated…
Question Number 552 by 123456 last updated on 25/Jan/15 $${proof}\:{that}\:\mathrm{log}_{\mathrm{2}} \mathrm{3}\:{is}\:{irrational} \\ $$ Answered by prakash jain last updated on 25/Jan/15 $$\mathrm{Let}\:\mathrm{us}\:\mathrm{say}\:\mathrm{log}_{\mathrm{2}} \mathrm{3}\:\mathrm{is}\:\mathrm{rational},\:\mathrm{then} \\ $$$$\mathrm{log}_{\mathrm{2}}…