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Category: Arithmetic

Question-192266

Question Number 192266 by Shlock last updated on 13/May/23 Answered by Frix last updated on 13/May/23 $$\mathrm{Since}\:{a}\in\mathbb{N}\wedge\mathrm{0}\leqslant{a}\leqslant\mathrm{9}\:\mathrm{it}'\mathrm{s}\:\mathrm{best}\:\mathrm{to}\:\mathrm{solve}\:\mathrm{for} \\ $$$${n}\:\mathrm{and}\:\mathrm{try}: \\ $$$$\mathrm{2}{n}^{\mathrm{2}} +\mathrm{14}{n}+\mathrm{83}=\mathrm{1111}{a} \\ $$$${n}=\frac{−\mathrm{7}+\sqrt{\mathrm{2222}{a}−\mathrm{117}}}{\mathrm{2}} \\…

Question-192208

Question Number 192208 by Shlock last updated on 11/May/23 Answered by witcher3 last updated on 11/May/23 $$\Rightarrow\forall\left(\mathrm{7k}+\mathrm{r}\right)\in\mathbb{N}\Rightarrow\exists\left(\mathrm{a},\mathrm{b},\mathrm{c}\right)\in\left[\mathrm{7k}+\mathrm{r},\mathrm{7k}+\mathrm{r}+\mathrm{6}\right] \\ $$$$\mathrm{r}\in\left\{\mathrm{0},\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4},\mathrm{5},\mathrm{6}\right\} \\ $$$$\mathrm{a}\equiv\mathrm{r}+\mathrm{1}\left[\mathrm{7}\right];\mathrm{a}=\mathrm{7k}+\mathrm{r}+\mathrm{1} \\ $$$$\mathrm{b}\equiv\mathrm{r}+\mathrm{2}\left[\mathrm{7}\right];\mathrm{b}=\mathrm{7k}+\mathrm{r}+\mathrm{2} \\ $$$$\mathrm{c}\equiv\mathrm{r}+\mathrm{4}\left[\mathrm{7}\right];\mathrm{c}=\mathrm{7k}+\mathrm{r}+\mathrm{4}…

z-1-z-2-a-z-2-z-3-b-z-3-z-4-c-z-1-z-4-d-solve-using-gaussian-elimination-

Question Number 192187 by alcohol last updated on 10/May/23 $$\begin{cases}{{z}_{\mathrm{1}} \:+\:{z}_{\mathrm{2}} \:=\:{a}}\\{{z}_{\mathrm{2}} \:+\:{z}_{\mathrm{3}} \:=\:{b}}\\{{z}_{\mathrm{3}} \:+\:{z}_{\mathrm{4}} \:=\:{c}}\\{{z}_{\mathrm{1}} \:+\:{z}_{\mathrm{4}} \:=\:{d}}\end{cases} \\ $$$${solve}\:{using}\:{gaussian}\:{elimination} \\ $$ Terms of Service…

a-n-N-a-n-1-pour-a-n-3-a-m-1modn-posons-m-n-1-subtituons-cette-valeur-dans-on-a-a-n-1-1modn-Mais-n-n-est-pas-forcement-premier-Test-de-primalite-n-N-n-3-n-2-

Question Number 61096 by Arthur El-bomart last updated on 29/May/19 $$\forall\:{a},\:{n}\:\in\:{N}\::\:\mid{a}−{n}\mid=\mathrm{1}\:{pour}\:{a},\:{n}\:\geqslant\mathrm{3} \\ $$$${a}^{{m}} \equiv\mathrm{1}{modn}\:\left(\ast\right) \\ $$$${posons}\::\:{m}={n}−\mathrm{1}\:\left(\ast'\right) \\ $$$${subtituons}\:{cette}\:{valeur}\:{dans}\:\left(\ast\right). \\ $$$${on}\:{a}:\:{a}^{{n}−\mathrm{1}} \equiv\mathrm{1}{modn}.\:{Mais}\:{n}\:{n}'{est}\:{pas}\:{forcement}\:{premier}. \\ $$$${Test}\:{de}\:{primalite} \\ $$$$\forall\:{n}\:\in\:{N},\:{n}\:\geqslant\mathrm{3}.…

5-12i-5-12i-

Question Number 60853 by Tony Lin last updated on 26/May/19 $$\sqrt{\mathrm{5}−\mathrm{12}{i}}+\sqrt{\mathrm{5}+\mathrm{12}{i}}=? \\ $$ Answered by tanmay last updated on 26/May/19 $$\sqrt{\mathrm{9}−\mathrm{4}−\mathrm{12}{i}}\:+\sqrt{\mathrm{9}−\mathrm{4}+\mathrm{12}{i}}\: \\ $$$$\sqrt{\left(\mathrm{3}−\mathrm{2}{i}\right)^{\mathrm{2}} }\:+\sqrt{\left(\mathrm{3}+\mathrm{2}{i}\right)^{\mathrm{2}} }\:…

Question-191831

Question Number 191831 by Shlock last updated on 01/May/23 Answered by mehdee42 last updated on 02/May/23 $${suppose}\:,\:\:{n}=<{abcdefghij}>\:,{is}\:{an}\:<{i}.{n}> \\ $$$${a}+{b}+{c}+…+{i}+{j}\overset{\mathrm{9}} {\equiv}\mathrm{0}\Rightarrow\mathrm{9}\mid{n} \\ $$$$\mathrm{9}\mid{n}\:,\:\mathrm{11111}\mid{n}\:\:,\:\left(\mathrm{9},\mathrm{11111}\right)=\mathrm{1}\Rightarrow\mathrm{99999}\mid{n} \\ $$$${let}\:\:{x}=<{abcde}>\:\:\&\:\:{y}={f}<{fghij}> \\…

Question-191758

Question Number 191758 by Mingma last updated on 30/Apr/23 Answered by AST last updated on 30/Apr/23 $$\mathrm{521}^{\mathrm{1000}} \equiv\mathrm{0001}\left({mod}\:\mathrm{10000}\right)\: \\ $$$$\left(\mathrm{521},\mathrm{10000}\right)=\mathrm{1}\:\Rightarrow\:\:\mathrm{521}^{\mathrm{999}} \equiv\frac{\mathrm{1}}{\mathrm{521}}\left({mod}\:\mathrm{10000}\right) \\ $$$$\frac{\mathrm{1}}{\mathrm{521}}\equiv{x}\left({mod}\mathrm{10000}\right)\Rightarrow\mathrm{521}{x}\equiv\mathrm{1}\left({mod}\:\mathrm{10000}\right) \\ $$$$\mathrm{10000}{q}−\mathrm{521}{x}=−\mathrm{1}…

If-a-sum-of-money-doubles-itself-in-a-time-T-when-compounded-continuously-find-the-rate-of-interest-in-terms-of-T-

Question Number 60441 by ajfour last updated on 21/May/19 $$\mathrm{If}\:\mathrm{a}\:\mathrm{sum}\:\mathrm{of}\:\:\mathrm{money}\:\mathrm{doubles}\:\mathrm{itself} \\ $$$$\mathrm{in}\:\mathrm{a}\:\mathrm{time}\:\mathrm{T},\:\mathrm{when}\:\mathrm{compounded} \\ $$$$\mathrm{continuously},\:\mathrm{find}\:\mathrm{the}\:\mathrm{rate}\:\mathrm{of} \\ $$$$\mathrm{interest},\:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{T}. \\ $$ Answered by tanmay last updated on 21/May/19…