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 $$\mathrm{Let}\:\mathrm{us}\:\mathrm{say}\:\mathrm{log}_{\mathrm{2}} \mathrm{3}\:\mathrm{is}\:\mathrm{rational},\:\mathrm{then} \\ $$$$\mathrm{log}_{\mathrm{2}}…
Question Number 548 by 123456 last updated on 25/Jan/15 $${how}\:{many}\:{digits}\:{are}\:{in}\:{periodic}\:{part} \\ $$$${of}\:\frac{\mathrm{1}}{\mathrm{60}^{\mathrm{30}} } \\ $$ Answered by prakash jain last updated on 25/Jan/15 $$\frac{\mathrm{1}}{\left(\mathrm{60}\right)^{\mathrm{30}} }=\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{30}}…
Question Number 518 by Yugi last updated on 25/Jan/15 $${Give}\:{the}\:{result}\:{of}\:{the}\:{following}\:{computation}\:{as}\:{an}\:{integer}\:{in}\:{the}\:{usual}\:{decimal} \\ $$$${form}.\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{303},\mathrm{000},\mathrm{000},\mathrm{000},\mathrm{303}×\mathrm{3},\mathrm{300},\mathrm{000},\mathrm{033}}{\mathrm{1},\mathrm{000},\mathrm{100},\mathrm{010},\mathrm{001}} \\ $$ Answered by prakash jain last updated on 22/Jan/15 $$\frac{\mathrm{303}\left(\mathrm{10}^{\mathrm{12}}…
Question Number 515 by 112358 last updated on 25/Jan/15 $${Find}\:{the}\:{smallest}\:{number}\:{greater} \\ $$$${than}\:{zero}\:{which}\:{can}\:{be}\:{written} \\ $$$${with}\:{ones}\:{and}\:{zeroes}\:{and}\:{is}\:{evenly}\:{divisble} \\ $$$${by}\:\mathrm{225}. \\ $$ Answered by prakash jain last updated on…
Question Number 514 by 112358 last updated on 25/Jan/15 $${Determine}\:{the}\:{smallest}\:{value}\:{of} \\ $$$${the}\:{form} \\ $$$${f}\left({u},{v}\right)=\frac{\mathrm{5}{v}^{\mathrm{2}} +\mathrm{5}{u}^{\mathrm{2}} +\mathrm{1}}{\mathrm{2}{u}+{v}} \\ $$$${where}\:{u},{v}\in{R}^{+} . \\ $$ Answered by prakash jain…