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Category: Number Theory

I-just-thought-of-something-I-am-curious-in-figuring-out-All-integer-numbers-can-be-made-up-by-prime-factors-That-is-n-p-1-p-2-p-i-n-Z-p-k-P-Are-there-an-inifinite-number-of-numbe

Question Number 3595 by Filup last updated on 16/Dec/15 $$\mathrm{I}\:\mathrm{just}\:\mathrm{thought}\:\mathrm{of}\:\mathrm{something}\:\mathrm{I}\:\mathrm{am}\:\mathrm{curious} \\ $$$$\mathrm{in}\:\mathrm{figuring}\:\mathrm{out}. \\ $$$$ \\ $$$$\mathrm{All}\:\mathrm{integer}\:\mathrm{numbers}\:\mathrm{can}\:\mathrm{be}\:\mathrm{made}\:\mathrm{up}\:\mathrm{by} \\ $$$${prime}\:{factors}.\:\mathrm{That}\:\mathrm{is}: \\ $$$${n}={p}_{\mathrm{1}} ×{p}_{\mathrm{2}} ×…×{p}_{{i}} \\ $$$${n}\in\mathbb{Z}\:\:\:\:\:\:\:\:\:{p}_{{k}} \in\mathbb{P}…

n-is-a-number-such-that-regular-n-gon-is-possible-with-straightedge-and-compass-only-Write-first-thirty-values-of-n-What-are-other-properties-of-such-numbers-If-values-of-n-are-arranged-in-or

Question Number 3519 by Rasheed Soomro last updated on 14/Dec/15 $${n}\:{is}\:{a}\:{number}\:{such}\:{that}\:{regular}\:\:{n}−{gon}\:{is} \\ $$$${possible}\:{with}\:{straightedge}\:{and}\:\:{compass}\:{only}. \\ $$$$\ast{Write}\:{first}\:{thirty}\:{values}\:{of}\:{n}. \\ $$$$\ast{What}\:{are}\:{other}\:{properties}\:{of}\:{such}\:{numbers}\:? \\ $$$$\ast{If}\:{values}\:{of}\:{n}\:{are}\:{arranged}\:{in}\:{order},\:{what}\:{is} \\ $$$${the}\:{formula}\:{for}\:{generating}\:{Nth}\:{number}? \\ $$ Commented by…

k-0-4-k-k-2-2k-1-2-2k-

Question Number 134393 by liberty last updated on 03/Mar/21 $$\:\underset{\mathrm{k}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\mathrm{4}^{\mathrm{k}} \:\left(\mathrm{k}!\right)^{\mathrm{2}} }{\left(\mathrm{2k}+\mathrm{1}\right)^{\mathrm{2}} \:\left(\mathrm{2k}\right)!}\:=? \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com

If-n-p-q-Z-and-p-and-q-are-coprimes-then-prove-that-HCF-of-n-p-1-and-n-q-1-is-n-1-Assume-n-gt-1-

Question Number 3296 by prakash jain last updated on 09/Dec/15 $$\mathrm{If}\:{n},{p},{q}\in\mathbb{Z}^{+} \:\mathrm{and}\:{p}\:\mathrm{and}\:{q}\:\mathrm{are}\:\mathrm{coprimes}, \\ $$$$\mathrm{then}\:\mathrm{prove}\:\mathrm{that}\:\mathrm{HCF}\:\mathrm{of}\:\left({n}^{{p}} −\mathrm{1}\right)\:\mathrm{and}\:\left({n}^{{q}} −\mathrm{1}\right)\:\mathrm{is}\:\left({n}−\mathrm{1}\right). \\ $$$$\mathrm{Assume}\:{n}>\mathrm{1}. \\ $$ Commented by Rasheed Soomro last…

Suggest-minimum-number-of-weights-two-peices-of-each-to-weigh-upto-at-least-60-kg-in-whole-kg-s-in-a-common-balance-

Question Number 3205 by Rasheed Soomro last updated on 07/Dec/15 $$\mathcal{S}{uggest}\:{minimum}\:{number}\:{of}\:\:{weights}\:,{two}\:{peices}\:{of}\:{each},\: \\ $$$${to}\:{weigh}\:{upto}\:{at}\:{least}\:\mathrm{60}\:{kg}\left({in}\:{whole}\:{kg}'{s}\right)\:{in}\:{a}\:{common} \\ $$$${balance}. \\ $$ Commented by prakash jain last updated on 07/Dec/15…

x-4-mod-5-x-3-mod-4-

Question Number 134267 by bramlexs22 last updated on 01/Mar/21 $$\begin{cases}{\mathrm{x}\equiv\:\mathrm{4}\:\left(\mathrm{mod}\:\mathrm{5}\right)}\\{\mathrm{x}\equiv\:\mathrm{3}\:\left(\mathrm{mod}\:\mathrm{4}\:\right)}\end{cases} \\ $$ Answered by EDWIN88 last updated on 01/Mar/21 $$\begin{cases}{\mathrm{x}=\mathrm{5k}+\mathrm{4}}\\{\mathrm{x}=\mathrm{4}\ell+\mathrm{3}}\end{cases}\:\Leftrightarrow\:\mathrm{5k}+\mathrm{4}\:=\:\mathrm{4}\ell+\mathrm{3} \\ $$$$\ell\:=\:\frac{\mathrm{5k}+\mathrm{1}}{\mathrm{4}}\:;\:\left(\mathrm{k},\ell\right)\:=\:\left(\mathrm{3},\mathrm{4}\right),\:\left(\mathrm{7},\mathrm{9}\right),\left(\mathrm{11},\mathrm{14}\right),…, \\ $$$$\left(\mathrm{4}\lambda−\mathrm{1},\:\mathrm{5}\lambda−\mathrm{1}\right)\:\Rightarrow\mathrm{then}\:\mathrm{x}=\mathrm{5}\left(\mathrm{4}\lambda−\mathrm{1}\right)+\mathrm{4} \\…

solve-x-2-mod-3-x-5-mod-7-

Question Number 134252 by bramlexs22 last updated on 01/Mar/21 $$\:\mathrm{solve}\:\begin{cases}{\mathrm{x}\equiv\:\mathrm{2}\:\left(\mathrm{mod}\:\mathrm{3}\right)}\\{\mathrm{x}\equiv\:\mathrm{5}\:\left(\mathrm{mod}\:\mathrm{7}\right)}\end{cases} \\ $$ Answered by EDWIN88 last updated on 01/Mar/21 $$\begin{cases}{\mathrm{x}\:=\:\mathrm{3k}+\mathrm{2}\:;\mathrm{k}\in\mathbb{Z}}\\{\mathrm{x}=\mathrm{7m}+\mathrm{5};\:\mathrm{m}\in\mathbb{Z}}\end{cases}\:\Rightarrow\:\mathrm{3k}+\mathrm{2}\:=\:\mathrm{7m}+\mathrm{5} \\ $$$$\Rightarrow\:\mathrm{m}=\frac{\mathrm{3k}−\mathrm{3}}{\mathrm{7}}\:;\:\left(\mathrm{k},\mathrm{m}\right)=\left(\mathrm{8},\mathrm{3}\right),\left(\mathrm{15},\mathrm{6}\right),\left(\mathrm{22},\mathrm{9}\right) \\ $$$$…,\:\left(\mathrm{7}\lambda+\mathrm{1},\:\mathrm{3}\lambda\right) \\…

How-many-different-clock-type-dials-can-be-made-containing-first-n-natual-numbers-with-the-property-that-sum-of-any-two-numbers-of-consecutive-positions-be-a-prime-number-N-1-2-3-

Question Number 3172 by Rasheed Soomro last updated on 07/Dec/15 $$\mathcal{H}{ow}\:{many}\:{different}\:\:{clock}−{type}\:{dials}\:{can}\:{be}\:{made}\: \\ $$$${containing}\:{first}\:{n}\:{natual}\:{numbers}\:{with}\:{the}\:{property} \\ $$$${that}\:{sum}\:{of}\:\:{any}\:{two}\:{numbers}\:{of}\:{consecutive}\:{positions}\:{be} \\ $$$${a}\:{prime}\:{number}. \\ $$$$\mathbb{N}=\left\{\mathrm{1},\mathrm{2},\mathrm{3},…\right\} \\ $$ Commented by Rasheed Soomro…