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} \\ $$$$\mathrm{triplet}\:\mathrm{for}\:\mathrm{all}\:\mathrm{m},\mathrm{n}\in\mathbb{N}.\:\mathrm{This}\:\mathrm{can}\:\mathrm{be} \\ $$$$\mathrm{proved}\:\mathrm{easily}.\mathrm{Is}\:\mathrm{the}\:\mathrm{vice}\:\mathrm{versa}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{statement}\:\mathrm{is}\:\mathrm{also}\:\mathrm{true}? \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{I}-\mathrm{E} \\ $$$$\mathrm{If}\:\:\mathrm{for}\:\mathrm{a},\mathrm{b},\mathrm{c}\in\mathbb{N}\:,\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}}…
Question Number 153916 by EDWIN88 last updated on 12/Sep/21 $${The}\:{value}\:{of}\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\left(\mathrm{3}_{{n}} \right)\left(\mathrm{2}_{{n}} \right){x}^{{n}} }{\left(\mathrm{1}_{{n}} \right){n}!}\:\beta\left(\mathrm{2},{n}+\mathrm{1}\right)\:{is} \\ $$$${a}.\:\frac{\mathrm{1}}{\mathrm{2}}\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left(\mathrm{2}_{{n}} \right)\frac{{x}^{{n}} }{{n}!} \\ $$$${b}.\:\frac{\mathrm{1}}{\mathrm{2}}\underset{{n}=\mathrm{0}} {\overset{\infty}…
Question Number 88261 by ar247 last updated on 09/Apr/20 Answered by Joel578 last updated on 09/Apr/20 $${u}\:\equiv\:{v}\:\left(\mathrm{mod}\:{m}\right)\:\Rightarrow\:{u}\:−\:{v}\:=\:{mx},\:{x}\:\in\:\mathbb{Z} \\ $$$$\left(\mathrm{1}\right)\:{u}\:=\:{mx}\:+\:{v} \\ $$$$\mathrm{Since}\:\mathrm{gcf}\left({v},{m}\right)\:\mathrm{divides}\:\mathrm{both}\:{v}\:\mathrm{and}\:\mathrm{m},\:\mathrm{it}\:\mathrm{also}\:\mathrm{divides}\:{u}, \\ $$$$\mathrm{hence}\:\mathrm{gcf}\left({v},{m}\right)\:\mathrm{divides}\:\mathrm{gcf}\left({u},{m}\right) \\ $$$$\Rightarrow\:\mathrm{gcf}\left({v},{m}\right)\:\mid\:\mathrm{gcf}\left({u},{m}\right)…
Question Number 22635 by Rasheed.Sindhi last updated on 21/Oct/17 $$\mathrm{For}\:\mathrm{each}\:\mathrm{positive}\:\mathrm{integer}\:\mathrm{n},\:\mathrm{define} \\ $$$$\mathrm{a}_{\mathrm{n}} =\mathrm{20}+\mathrm{n}^{\mathrm{2}} ,\mathrm{and}\:\mathrm{d}_{\mathrm{n}} =\mathrm{gcd}\left(\mathrm{a}_{\mathrm{n}} ,\mathrm{a}_{\mathrm{n}+\mathrm{2}} \right). \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{all}\:\mathrm{values}\:\mathrm{that}\:\mathrm{are} \\ $$$$\mathrm{taken}\:\mathrm{by}\:\mathrm{d}_{\mathrm{n}} . \\ $$$$ \\…
Question Number 22625 by Rasheed.Sindhi last updated on 21/Oct/17 $$\mathrm{For}\:\mathrm{each}\:\mathrm{positive}\:\mathrm{integer}\:\mathrm{n}\:\mathrm{define} \\ $$$$\mathrm{a}_{\mathrm{n}} =\mathrm{30}+\mathrm{n}^{\mathrm{2}} ,\mathrm{and}\:\mathrm{d}_{\mathrm{n}} =\mathrm{gcd}\left(\mathrm{a}_{\mathrm{n}} ,\mathrm{a}_{\mathrm{n}+\mathrm{1}} \right). \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{all}\:\mathrm{values}\:\mathrm{that}\:\mathrm{are} \\ $$$$\mathrm{taken}\:\mathrm{by}\:\mathrm{d}_{\mathrm{n}} \:\mathrm{and}\:\mathrm{show}\:\mathrm{by}\:\mathrm{examples} \\ $$$$\mathrm{that}\:\mathrm{each}\:\mathrm{of}\:\mathrm{these}\:\mathrm{values}\:\mathrm{are}\:\mathrm{attained}. \\…