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Category: Number Theory

Find-the-cube-root-of-one-Hence-show-that-the-sum-of-the-root-is-equal-to-zero-

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}}…

A-four-digit-whole-number-is-interesting-if-the-number-formed-by-the-leftmost-two-digits-is-twice-as-large-as-the-number-formed-by-the-rightmost-two-digits-for-example-2010-is-interesting-1-find-t

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}} \\…

Find-the-minimum-possible-least-common-multiple-lcm-of-twenty-not-necessarily-distinct-natural-numbers-whose-sum-is-801-

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…

n-0-1-n-2n-1-3-

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}}…

how-many-positive-x-10-000-integers-are-such-that-2-x-x-2-is-divisible-by-7-

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…

Prove-that-3k-2-is-not-perfect-square-for-all-k-0-1-2-3-

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}, \\…

m-2-n-2-2mn-m-2-n-2-is-pythagorean-triplet-for-all-m-n-N-This-can-be-proved-easily-Is-the-vice-versa-of-the-statement-is-also-true-I-E-If-for-a-b-c-N-a-2-b

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}}…

The-value-of-n-0-3-n-2-n-x-n-1-n-n-2-n-1-is-a-1-2-n-0-2-n-x-n-n-b-1-2-n-0-3-n-2-n-1-n-x-n-n-c-1-2-n-0-2-n-x-n-1-

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-88261

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)…