Question Number 15787 by tawa tawa last updated on 13/Jun/17 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{four}\:\mathrm{digits}\:\mathrm{number}\:\mathrm{such}\:\mathrm{that}\:\:\mathrm{4}\:\centerdot\:\mathrm{abcd}\:=\:\mathrm{dcba} \\ $$ Commented by mrW1 last updated on 13/Jun/17 $$\mathrm{I}\:\mathrm{found}\:\mathrm{only}\:\mathrm{4}×\mathrm{2178}=\mathrm{8712} \\ $$ Commented by…
Question Number 81318 by ahmadshahhimat775@gmail.com last updated on 11/Feb/20 Commented by mr W last updated on 11/Feb/20 $$\mathrm{8}^{\mathrm{2}{n}} =\left(\mathrm{9}−\mathrm{1}\right)^{\mathrm{2}{n}} \equiv\mathrm{1}\:{mod}\:\mathrm{9} \\ $$ Terms of Service…
Question Number 15742 by RasheedSoomro last updated on 14/Jun/17 $$\mathrm{This}\:\mathrm{question}\:\mathrm{is}\:\mathrm{posted}\:\mathrm{on}\:\mathrm{the}\:\mathrm{request}\:\mathrm{of}\:\mathrm{mrW1}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\left(\mathrm{See}\:\mathrm{comments}\:\mathrm{of}\:\mathrm{my}\:\mathrm{answer}\:\mathrm{to}\:\mathrm{Q}#\mathrm{15543}\right). \\ $$$$ \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{last}\:\mathrm{last}\:\:\boldsymbol{\mathrm{non}}-\boldsymbol{\mathrm{zero}}\:\mathrm{digit}\:\mathrm{of}\:\mathrm{the}\:\mathrm{expansion} \\ $$$$\mathrm{of}\:\:\mathrm{2000}! \\ $$ Commented by mrW1 last updated…
Question Number 15543 by RasheedSoomro last updated on 11/Jun/17 $$\mathrm{Q}#\mathrm{13724}\:\mathcal{R}{eposted}. \\ $$$$\mathcal{E}_{\:} ^{\:} \mathrm{xpansion}\:\mathrm{of}\:\mathrm{1000}! \\ $$$$\mathrm{has}\:\mathrm{24}\:\:\mathrm{0}'\mathrm{s}\:\mathrm{at}\:\mathrm{the}\:\mathrm{end}.\:\mathrm{Find} \\ $$$$\mathrm{the}\:\mathrm{first}\:\mathrm{non}-\:\mathrm{zero}\:\mathrm{digit}\: \\ $$$$\mathrm{from}\:\mathrm{right}. \\ $$$$\mathrm{1000}!=….\mathrm{d0000}…\mathrm{0} \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{d}? \\…
Question Number 15377 by Joel577 last updated on 10/Jun/17 $$−\mathrm{39}\:\mathrm{mod}\:\mathrm{4}\:=\:? \\ $$ Answered by Tinkutara last updated on 10/Jun/17 $$−\mathrm{39}\:\equiv\:−\mathrm{3}\:\left(\mathrm{mod}\:\mathrm{4}\right) \\ $$$$\equiv\:\mathrm{1}\:\left(\mathrm{mod}\:\mathrm{4}\right) \\ $$$$\mathrm{You}\:\mathrm{can}\:\mathrm{check}:\:−\mathrm{39}\:=\:\mathrm{4}\left(−\mathrm{10}\right)\:+\:\mathrm{1} \\…
Question Number 15322 by tawa tawa last updated on 09/Jun/17 Commented by mrW1 last updated on 09/Jun/17 $$\mathrm{sin}^{\mathrm{2}} \:\mathrm{x}_{\mathrm{k}} =\frac{\mathrm{4}}{\mathrm{2015}} \\ $$$$\mathrm{cos}^{\mathrm{2}} \:\mathrm{x}_{\mathrm{k}} =\mathrm{1}−\frac{\mathrm{4}}{\mathrm{2015}}=\frac{\mathrm{2011}}{\mathrm{2015}} \\…
Question Number 15292 by RasheedSoomro last updated on 09/Jun/17 $$\mathrm{Light}\:\mathrm{version}\:\mathrm{of}\:\mathcal{Q}#\mathrm{13724} \\ $$$$\mathcal{E}\mathrm{xpansion}\:\mathrm{of}\:\mathrm{100}!\:\mathrm{has}\:\mathrm{24},\:\mathrm{0}'\mathrm{s}\:\:\mathrm{at}\:\mathrm{the}\:\mathrm{end}. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{first}\:\mathrm{non}-\mathrm{zero}\:\mathrm{digit}\:\mathrm{from}\:\mathrm{right}. \\ $$$$\mathrm{100}!=…..\mathrm{d000}…\mathrm{00} \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{d}? \\ $$ Answered by RasheedSoomro last updated…
Question Number 14949 by Tinkutara last updated on 05/Jun/17 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{largest}\:\mathrm{prime}\:\mathrm{factor}\:\mathrm{of}\:\mathrm{203203}. \\ $$$$\mathrm{Anyone}\:\mathrm{please}\:\mathrm{suggest}\:\mathrm{the}\:\mathrm{method} \\ $$$$\mathrm{without}\:\mathrm{calculators}\:\mathrm{or}\:\mathrm{log}\:\mathrm{tables}. \\ $$ Commented by RasheedSoomro last updated on 05/Jun/17 $$\mathrm{203203}=\mathrm{203000}+\mathrm{203}=\mathrm{203}\left(\mathrm{1000}+\mathrm{1}\right) \\…
Question Number 14757 by RasheedSoomro last updated on 04/Jun/17 $$\mathrm{Solve}: \\ $$$$\frac{\mathrm{1000}!}{\mathrm{5}×\mathrm{10}×\mathrm{15}×…\mathrm{1000}}\equiv\mathrm{x}\left(\mathrm{mod}\:\mathrm{10}\right) \\ $$ Answered by RasheedSoomro last updated on 05/Jun/17 $$\mathrm{x}\equiv\frac{\mathrm{1000}!}{\mathrm{5}×\mathrm{10}×\mathrm{15}×…\mathrm{1000}}\left(\mathrm{mod}\:\mathrm{10}\right. \\ $$$$\mathrm{x}\equiv\left(\mathrm{1}.\mathrm{2}.\mathrm{3}.\mathrm{4}\right)\left(\mathrm{6}.\mathrm{7}.\mathrm{8}.\mathrm{9}\right)…\left(\mathrm{996}.\mathrm{997}.\mathrm{998}.\mathrm{999}\right)\left(\mathrm{mod}\:\mathrm{10}\right) \\…
Question Number 14715 by tawa tawa last updated on 03/Jun/17 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{when}\:\:\:\mathrm{55}^{\mathrm{99}} \:\:\mathrm{is}\:\mathrm{divided}\:\mathrm{by}\:\:\mathrm{14} \\ $$ Commented by prakash jain last updated on 04/Jun/17 $$\mathrm{55}\:\mathrm{mod}\:\mathrm{14}=−\mathrm{1} \\ $$$$\mathrm{55}^{\mathrm{99}}…