Question Number 24944 by soufiane zarik last updated on 29/Nov/17 Commented by prakash jain last updated on 30/Nov/17 $${x}={n}+{f}\:\:\mathrm{0}\leqslant{f}<\mathrm{1},\:{n}\in\mathbb{Z},{n}\geqslant\mathrm{0} \\ $$$$\left[\left({n}+{f}\right)^{\mathrm{2}} \right]=\left[{n}+{f}\right]^{\mathrm{2}} \\ $$$$\Rightarrow\left[{n}^{\mathrm{2}} +\mathrm{2}{nf}+{f}^{\mathrm{2}}…
Question Number 90470 by Asif Hypothesis last updated on 23/Apr/20 $${For}\:{any}\:{positive}\:{integer}\:{n},\:\tau\left({n}\right)\:{is}\:{the}\:{number}\:{of}\:{its}\:{factors}\: \\ $$$${Prove}, \\ $$$$\sum_{{i}=\mathrm{1}} ^{{n}} \tau\left({i}\right)=\sum_{{i}=\mathrm{1}} ^{{n}} \lfloor{n}/{i}\rfloor \\ $$ Terms of Service Privacy…
Question Number 155571 by peter frank last updated on 02/Oct/21 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{cube}\:\mathrm{root}\:\mathrm{of}\:\mathrm{one}\:.\mathrm{Hence} \\ $$$$\mathrm{show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{root}\:\mathrm{is}\: \\ $$$$\mathrm{equal}\:\mathrm{to}\:\mathrm{zero} \\ $$ Answered by JDamian last updated on 02/Oct/21 $${z}^{\mathrm{3}}…
Question Number 89878 by malwaan last updated on 19/Apr/20 $$\boldsymbol{{A}}\:\boldsymbol{{four\_digit}}\:\boldsymbol{{whole}}\:\boldsymbol{{number}} \\ $$$$\boldsymbol{{is}}\:\boldsymbol{{interesting}}\:\boldsymbol{{if}}\:\boldsymbol{{the}}\:\boldsymbol{{number}} \\ $$$$\boldsymbol{{formed}}\:\boldsymbol{{by}}\:\boldsymbol{{the}}\:\boldsymbol{{leftmost}}\:\boldsymbol{{two}} \\ $$$$\boldsymbol{{digits}}\:\boldsymbol{{is}}\:\boldsymbol{{twice}}\:\boldsymbol{{as}}\:\boldsymbol{{large}}\:\boldsymbol{{as}}\:\boldsymbol{{the}} \\ $$$$\boldsymbol{{number}}\:\boldsymbol{{formed}}\:\boldsymbol{{by}}\:\boldsymbol{{the}} \\ $$$$\boldsymbol{{rightmost}}\:\boldsymbol{{two}}\:\boldsymbol{{digits}}. \\ $$$$\left(\boldsymbol{{for}}\:\boldsymbol{{example}}\:\mathrm{2010}\:\boldsymbol{{is}}\:\boldsymbol{{interesting}}\right) \\ $$$$\mathrm{1}\:\:\boldsymbol{{find}}\:\boldsymbol{{the}}\:\boldsymbol{{largest}}\:\boldsymbol{{whole}}\:\boldsymbol{{number}} \\…
Question Number 24324 by Tinkutara last updated on 15/Nov/17 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{possible}\:\mathrm{least} \\ $$$$\mathrm{common}\:\mathrm{multiple}\:\left(\mathrm{lcm}\right)\:\mathrm{of}\:\mathrm{twenty}\:\left(\mathrm{not}\right. \\ $$$$\left.\mathrm{necessarily}\:\mathrm{distinct}\right)\:\mathrm{natural}\:\mathrm{numbers} \\ $$$$\mathrm{whose}\:\mathrm{sum}\:\mathrm{is}\:\mathrm{801}. \\ $$ Commented by Tinkutara last updated on 16/Nov/17…
Question Number 155133 by amin96 last updated on 25/Sep/21 $$\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)^{\mathrm{3}} }=? \\ $$ Answered by mnjuly1970 last updated on 26/Sep/21 $$\:\:\:\frac{\pi^{\:\mathrm{3}} }{\mathrm{32}}…
Question Number 23612 by Joel577 last updated on 02/Nov/17 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{last}\:\mathrm{2}\:\mathrm{digits}\:\mathrm{from} \\ $$$$\mathrm{20}^{\mathrm{17}} \:+\:\mathrm{17}^{\mathrm{20}} \\ $$ Commented by Rasheed.Sindhi last updated on 02/Nov/17 $$\mathrm{20}^{\mathrm{17}} =\left(\mathrm{2}×\mathrm{10}\right)^{\mathrm{17}} =\mathrm{2}^{\mathrm{17}}…
Question Number 154672 by talminator2856791 last updated on 20/Sep/21 $$\: \\ $$$$\:\mathrm{how}\:\mathrm{many}\:\mathrm{positive}\:{x}\leqslant\mathrm{10}\:\mathrm{000}\:\mathrm{integers}\:\mathrm{are}\:\: \\ $$$$\:\mathrm{such}\:\mathrm{that}\:\mathrm{2}^{{x}} −{x}^{\mathrm{2}} \:\mathrm{is}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{7}? \\ $$$$\: \\ $$ Answered by MJS_new last updated…
Question Number 23269 by Rasheed.Sindhi last updated on 28/Oct/17 $$\mathbb{P}\mathrm{rove}\:\mathrm{that} \\ $$$$\:\mathrm{3k}+\mathrm{2}\:\mathrm{is}\:\mathrm{not}\:\mathrm{perfect}\:\mathrm{square}\:\mathrm{for} \\ $$$$\mathrm{all}\:\mathrm{k}\in\left\{\mathrm{0},\mathrm{1},\mathrm{2},\mathrm{3},…\right\}. \\ $$ Answered by Tinkutara last updated on 28/Oct/17 $${Any}\:{number}\:{can}\:{be}\:{written}\:{as}\:\mathrm{3}{p}, \\…
Question Number 23105 by Rasheed.Sindhi last updated on 26/Oct/17 $$\left(\mid\mathrm{m}^{\mathrm{2}} −\mathrm{n}^{\mathrm{2}} \mid,\mathrm{2mn},\mathrm{m}^{\mathrm{2}} +\mathrm{n}^{\mathrm{2}} \right)\:\mathrm{is}\:\mathrm{pythagorean} \\ $$$$\mathrm{triplet}\:\mathrm{for}\:\mathrm{all}\:\mathrm{m},\mathrm{n}\in\mathbb{N}.\:\mathrm{This}\:\mathrm{can}\:\mathrm{be} \\ $$$$\mathrm{proved}\:\mathrm{easily}.\mathrm{Is}\:\mathrm{the}\:\mathrm{vice}\:\mathrm{versa}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{statement}\:\mathrm{is}\:\mathrm{also}\:\mathrm{true}? \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{I}-\mathrm{E} \\ $$$$\mathrm{If}\:\:\mathrm{for}\:\mathrm{a},\mathrm{b},\mathrm{c}\in\mathbb{N}\:,\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}}…