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Question Number 166911 Answers: 1 Comments: 0

Question Number 166910 Answers: 1 Comments: 1

given that is prime,proof that (√p) is irrational

$${given}\:{that}\:{is}\:{prime},{proof}\:{that}\:\sqrt{{p}}\:{is}\: \\ $$$${irrational} \\ $$

Question Number 166881 Answers: 0 Comments: 0

Question Number 166602 Answers: 0 Comments: 0

Determine condition/s that a number divisible by 11 is a palindrome-number.

$$\mathcal{D}{etermine}\:{condition}/{s}\:{that} \\ $$$${a}\:\boldsymbol{{number}}\:\boldsymbol{{divisible}}\:\boldsymbol{{by}}\:\mathrm{11} \\ $$$${is}\:{a}\:\boldsymbol{{palindrome}}-\boldsymbol{{number}}. \\ $$

Question Number 166601 Answers: 0 Comments: 0

What is the condition that a palindrome-number is divisible by 11?

$${What}\:{is}\:{the}\:{condition}\:{that}\: \\ $$$${a}\:\boldsymbol{{palindrome}}-\boldsymbol{{number}} \\ $$$$\:{is}\:\boldsymbol{{divisible}}\:\boldsymbol{{by}}\:\mathrm{11}? \\ $$

Question Number 166530 Answers: 1 Comments: 0

Question Number 166475 Answers: 2 Comments: 5

Find out n∈N such that n^2 +n is divisible by 30.

$$\mathcal{F}{ind}\:{out}\:{n}\in\mathbb{N} \\ $$$$\:{such}\:{that} \\ $$$${n}^{\mathrm{2}} +{n}\:{is}\:{divisible}\:{by}\:\mathrm{30}. \\ $$

Question Number 164940 Answers: 0 Comments: 1

−−−−−−−−− 1!−2!+3!−4!+5!−…−14!+15!=? −−−−−−−−−−by M.A

$$−−−−−−−−− \\ $$$$\mathrm{1}!−\mathrm{2}!+\mathrm{3}!−\mathrm{4}!+\mathrm{5}!−\ldots−\mathrm{14}!+\mathrm{15}!=? \\ $$$$ \\ $$$$−−−−−−−−−−\boldsymbol{{by}}\:\boldsymbol{{M}}.\boldsymbol{{A}} \\ $$

Question Number 164279 Answers: 3 Comments: 0

x^n =n^x x=?

$${x}^{{n}} ={n}^{{x}} \:\:\:{x}=? \\ $$

Question Number 163763 Answers: 3 Comments: 1

$$ \\ $$

Question Number 163662 Answers: 1 Comments: 1

Question Number 163397 Answers: 0 Comments: 5

Question Number 163300 Answers: 1 Comments: 0

∫_0 ^2 f(x)dx⋍f(a)+f(2−a) For what values of a is the following formula accurate for polynomials of degree 3?

$$\int_{\mathrm{0}} ^{\mathrm{2}} \boldsymbol{{f}}\left(\boldsymbol{{x}}\right)\boldsymbol{{dx}}\backsimeq\boldsymbol{{f}}\left(\boldsymbol{{a}}\right)+\boldsymbol{{f}}\left(\mathrm{2}−\boldsymbol{{a}}\right) \\ $$$$ \\ $$For what values of a is the following formula accurate for polynomials of degree 3?

Question Number 163168 Answers: 2 Comments: 1

Question Number 162674 Answers: 2 Comments: 0

Question Number 162169 Answers: 2 Comments: 0

determinant ((( determinant ((( x+y+x^2 y^2 =586_(x=?,y=? ) ^(x,y∈Z ) )))_ ^ _() ^(•) )))

$$ \\ $$$$\:\:\:\:\:\:\:\begin{array}{|c|}{\overset{\bullet} {\:\:\:\:\:\begin{array}{|c|}{\:\:\:\underset{{x}=?,{y}=?\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:} {\overset{{x},{y}\in\mathbb{Z}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:} {{x}+{y}+{x}^{\mathrm{2}} {y}^{\mathrm{2}} =\mathrm{586}}}\:\:}\\\hline\end{array}_{} ^{} }\:\:\:\:}\\\hline\end{array} \\ $$$$ \\ $$

Question Number 161860 Answers: 1 Comments: 0

Simplify ((1^2 ∙2!+2^2 ∙3!+3^2 ∙4!+∙∙∙+n^2 (n+1)!−2)/((n+1)!)) to n^2 +n−2

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Simplify} \\ $$$$\frac{\mathrm{1}^{\mathrm{2}} \centerdot\mathrm{2}!+\mathrm{2}^{\mathrm{2}} \centerdot\mathrm{3}!+\mathrm{3}^{\mathrm{2}} \centerdot\mathrm{4}!+\centerdot\centerdot\centerdot+{n}^{\mathrm{2}} \left({n}+\mathrm{1}\right)!−\mathrm{2}}{\left({n}+\mathrm{1}\right)!} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{to} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{n}^{\mathrm{2}} +\mathrm{n}−\mathrm{2} \\ $$

Question Number 161843 Answers: 0 Comments: 3

Q#161744 reposted with some change. Solve for integer numbers: (x/y) + (5/x) + ((y - 5)/5) = ((y + x)/(y + 5)) + ((5 + y)/(5 + x))

$${Q}#\mathrm{161744}\:{reposted}\:{with}\:{some}\:{change}. \\ $$$$\mathrm{Solve}\:\mathrm{for}\:\boldsymbol{\mathrm{integer}}\:\mathrm{numbers}: \\ $$$$\frac{\mathrm{x}}{\mathrm{y}}\:+\:\frac{\mathrm{5}}{\mathrm{x}}\:+\:\frac{\mathrm{y}\:-\:\mathrm{5}}{\mathrm{5}}\:=\:\frac{\mathrm{y}\:+\:\mathrm{x}}{\mathrm{y}\:+\:\mathrm{5}}\:+\:\frac{\mathrm{5}\:+\:\mathrm{y}}{\mathrm{5}\:+\:\mathrm{x}} \\ $$

Question Number 161622 Answers: 1 Comments: 0

Σ_(n=1) ^∞ (((−1)^(n+1) )/(n(2n+1)))=?

$$\underset{\boldsymbol{{n}}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{\boldsymbol{{n}}+\mathrm{1}} }{\boldsymbol{{n}}\left(\mathrm{2}\boldsymbol{{n}}+\mathrm{1}\right)}=? \\ $$

Question Number 161528 Answers: 1 Comments: 0

PROVE that the numbers of types 4k+2 & 4k+3 are NOT perfect □s.

$$\mathrm{PROVE}\:\mathrm{that}\:\mathrm{the}\:\mathrm{numbers}\:\mathrm{of}\:\mathrm{types} \\ $$$$\mathrm{4k}+\mathrm{2}\:\&\:\mathrm{4k}+\mathrm{3}\:\mathrm{are}\:\mathrm{NOT}\:\mathrm{perfect}\:\Box\mathrm{s}. \\ $$

Question Number 160270 Answers: 2 Comments: 0

Question Number 160061 Answers: 1 Comments: 0

Find out some pairs (a,b) such that for some n≥1 a^n +b^n ,a^(2n) +b^(2n) ,a^(4n) +b^(4n) ,a^(8n) +b^(8n) ∈P

$$ \\ $$$${Find}\:{out}\:{some}\:{pairs}\:\left({a},{b}\right)\:{such}\:{that} \\ $$$${for}\:{some}\:{n}\geqslant\mathrm{1} \\ $$$${a}^{{n}} +{b}^{{n}} ,{a}^{\mathrm{2}{n}} +{b}^{\mathrm{2}{n}} ,{a}^{\mathrm{4}{n}} +{b}^{\mathrm{4}{n}} ,{a}^{\mathrm{8}{n}} +{b}^{\mathrm{8}{n}} \in\mathbb{P} \\ $$$$ \\ $$

Question Number 159925 Answers: 1 Comments: 0

Find out pairs of numbers (a,b) (as many as you can) such that: (√a) +(√b) , a+b , a^2 +b^2 ∈ P

$$\mathrm{Find}\:\mathrm{out}\:\mathrm{pairs}\:\mathrm{of}\:\mathrm{numbers}\:\left(\mathrm{a},\mathrm{b}\right)\:\left(\mathrm{as}\right. \\ $$$$\left.\mathrm{many}\:\mathrm{as}\:\mathrm{you}\:\mathrm{can}\right)\:\mathrm{such}\:\mathrm{that}: \\ $$$$\sqrt{\mathrm{a}}\:+\sqrt{\mathrm{b}}\:,\:\mathrm{a}+\mathrm{b}\:,\:\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} \:\in\:\mathbb{P} \\ $$

Question Number 159775 Answers: 1 Comments: 0

Π_(n=1) ^∞ ((α^3 +β^2 )/3^n )= ? in expanded form

$$\prod_{\mathrm{n}=\mathrm{1}} ^{\infty} \frac{\alpha^{\mathrm{3}} +\beta^{\mathrm{2}} }{\mathrm{3}^{\mathrm{n}} }=\:? \\ $$$$\mathrm{in}\:\mathrm{expanded}\:\mathrm{form} \\ $$

Question Number 159461 Answers: 0 Comments: 4

≺PRIME-BIRTHDAYS≻ Do you know ′Prime1611′?... No,no it′s not an ID of the forum- member.It is a person who was born on November 16, 0001.On his birthday astrologers formed a string from his birthdate in the way: ′ddmmyyyy′: ′16110001′ The astro- logers observed that number containing in the string: 16110001 is a prime number.Thus they gave him name ′Prime1611′ They also suggested that Mr Prime1611 should celebrate his birthday only when the number ddmmyyyy be prime number.Recently He celebrated his birthday on 16-11-2021 as the number 16112021 is prime. How many birthdays has he celebrated upto his recent birthday when he had celebrated his first birthday on 16-11-0001? You may use calculator also. Question connected with Q#159421

$$\:\:\:\:\:\:\:\:\:\:\prec\mathbb{PRIME}-\mathcal{BIRTHDAYS}\succ \\ $$$$\mathrm{Do}\:\mathrm{you}\:\mathrm{know}\:'\mathrm{Prime1611}'?... \\ $$$$\mathrm{No},\mathrm{no}\:\mathrm{it}'\mathrm{s}\:\mathrm{not}\:\mathrm{an}\:\mathrm{ID}\:\mathrm{of}\:\mathrm{the}\:\mathrm{forum}- \\ $$$$\mathrm{member}.\mathrm{It}\:\mathrm{is}\:\mathrm{a}\:\mathrm{person}\:\mathrm{who}\:\mathrm{was}\:\mathrm{born} \\ $$$$\mathrm{on}\:\mathrm{November}\:\mathrm{16},\:\mathrm{0001}.\mathrm{On}\:\mathrm{his} \\ $$$$\mathrm{birthday}\:\mathrm{astrologers}\:\mathrm{formed}\:\mathrm{a}\:\mathrm{string} \\ $$$$\mathrm{from}\:\mathrm{his}\:\mathrm{birthdate}\:\mathrm{in}\:\mathrm{the}\:\mathrm{way}: \\ $$$$'\mathrm{ddmmyyyy}':\:'\mathrm{16110001}'\:\mathrm{The}\:\mathrm{astro}- \\ $$$$\mathrm{logers}\:\mathrm{observed}\:\mathrm{that}\:\mathrm{number}\:\mathrm{containing} \\ $$$$\mathrm{in}\:\mathrm{the}\:\mathrm{string}:\:\mathrm{16110001}\:\mathrm{is}\:\mathrm{a}\:\mathrm{prime} \\ $$$$\mathrm{number}.\mathrm{Thus}\:\mathrm{they}\:\mathrm{gave}\:\mathrm{him}\:\mathrm{name} \\ $$$$'\mathrm{Prime1611}'\:\mathrm{They}\:\mathrm{also}\:\mathrm{suggested}\:\mathrm{that} \\ $$$$\mathrm{Mr}\:\mathrm{Prime1611}\:\mathrm{should}\:\mathrm{celebrate}\:\mathrm{his} \\ $$$$\mathrm{birthday}\:\mathrm{only}\:\mathrm{when}\:\mathrm{the}\:\mathrm{number} \\ $$$$\mathrm{ddmmyyyy}\:\mathrm{be}\:\mathrm{prime}\:\mathrm{number}.\mathrm{Recently} \\ $$$$\mathrm{He}\:\mathrm{celebrated}\:\mathrm{his}\:\mathrm{birthday}\:\mathrm{on} \\ $$$$\mathrm{16}-\mathrm{11}-\mathrm{2021}\:\mathrm{as}\:\mathrm{the}\:\mathrm{number}\:\mathrm{16112021} \\ $$$$\mathrm{is}\:\mathrm{prime}. \\ $$$$\mathrm{How}\:\mathrm{many}\:\mathrm{birthdays}\:\mathrm{has}\:\mathrm{he}\:\mathrm{celebrated} \\ $$$$\mathrm{upto}\:\mathrm{his}\:\mathrm{recent}\:\mathrm{birthday}\:\mathrm{when}\:\mathrm{he} \\ $$$$\mathrm{had}\:\mathrm{celebrated}\:\mathrm{his}\:\mathrm{first}\:\mathrm{birthday}\:\mathrm{on} \\ $$$$\mathrm{16}-\mathrm{11}-\mathrm{0001}? \\ $$$$\mathrm{You}\:\mathrm{may}\:\mathrm{use}\:\mathrm{calculator}\:\mathrm{also}. \\ $$$$\mathrm{Question}\:\mathrm{connected}\:\mathrm{with}\:\mathrm{Q}#\mathrm{159421} \\ $$

Question Number 158914 Answers: 0 Comments: 0

Σ_(n=0) ^∞ (1/((3n+1)^3 ))=?

$$\underset{\boldsymbol{\mathrm{n}}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\left(\mathrm{3}\boldsymbol{\mathrm{n}}+\mathrm{1}\right)^{\mathrm{3}} }=? \\ $$

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