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Question Number 217129 Answers: 1 Comments: 2

prove that if an integer n is not divisible by 2 or 3 then n^2 ≡1(mod 24)

$${prove}\:{that}\:{if}\:{an}\:{integer}\:{n}\:{is}\:{not}\:{divisible}\:{by}\:\mathrm{2}\:{or}\:\mathrm{3} \\ $$$$\:{then}\:{n}^{\mathrm{2}} \equiv\mathrm{1}\left({mod}\:\mathrm{24}\right) \\ $$

Question Number 217088 Answers: 1 Comments: 0

show that ∫_( n) ^( n + 1) ln(t) dt ≤ ln(n + (1/2)) Given u_n = (((4n)^n n!e^(−n) )/((2n)!)), ∀n ≥ 1 prove, using the preceding question that u_n is decreasing and convergent

$${show}\:{that}\:\int_{\:{n}} ^{\:{n}\:+\:\mathrm{1}} {ln}\left({t}\right)\:{dt}\:\leqslant\:{ln}\left({n}\:+\:\frac{\mathrm{1}}{\mathrm{2}}\right) \\ $$$${Given}\:{u}_{{n}} \:=\:\frac{\left(\mathrm{4}{n}\right)^{{n}} {n}!{e}^{−{n}} }{\left(\mathrm{2}{n}\right)!},\:\forall{n}\:\geqslant\:\mathrm{1} \\ $$$${prove},\:{using}\:{the}\:{preceding}\:{question}\:{that} \\ $$$${u}_{{n}} \:{is}\:{decreasing}\:{and}\:{convergent} \\ $$

Question Number 217085 Answers: 0 Comments: 2

Question Number 217084 Answers: 0 Comments: 0

Geometrie dans le plan. AB^(→) et CD^(→) sont deux vecteurs du plan. AB^(→) n′est pas nul. Demontre que si AB^(→) et CD^(→) sont colineaires alors il existe un nombre reel k tel que CD^(→) = k AB^(→) .

$$\boldsymbol{\mathrm{Geometr}}\mathrm{i}\boldsymbol{\mathrm{e}}\:\boldsymbol{\mathrm{dans}}\:\boldsymbol{\mathrm{le}}\:\boldsymbol{\mathrm{plan}}. \\ $$$$\overset{\rightarrow} {\mathrm{AB}}\:\mathrm{et}\:\overset{\rightarrow} {\mathrm{CD}}\:\mathrm{sont}\:\mathrm{deux}\:\mathrm{vecteurs}\:\mathrm{du}\:\mathrm{plan}. \\ $$$$\overset{\rightarrow} {\mathrm{AB}}\:\mathrm{n}'\mathrm{est}\:\mathrm{pas}\:\mathrm{nul}. \\ $$$$\mathrm{Demontre}\:\mathrm{que}\:\mathrm{si}\:\overset{\rightarrow} {\mathrm{AB}}\:\mathrm{et}\:\overset{\rightarrow} {\mathrm{CD}}\:\mathrm{sont}\:\mathrm{colineaires} \\ $$$$\mathrm{alors}\:\mathrm{il}\:\mathrm{existe}\:\mathrm{un}\:\mathrm{nombre}\:\mathrm{reel}\:\mathrm{k}\:\mathrm{tel}\:\mathrm{que} \\ $$$$\overset{\rightarrow} {\mathrm{CD}}\:=\:\mathrm{k}\:\overset{\rightarrow} {\mathrm{AB}}. \\ $$

Question Number 217080 Answers: 3 Comments: 1

Question Number 217068 Answers: 0 Comments: 0

Question Number 217066 Answers: 1 Comments: 0

Find all integer x,y such that x^2 −y^2 =100

$${Find}\:{all}\:{integer}\:{x},{y}\:{such}\:{that} \\ $$$${x}^{\mathrm{2}} −{y}^{\mathrm{2}} =\mathrm{100} \\ $$

Question Number 217065 Answers: 0 Comments: 9

To Tinku Tara sir, How can I import something in Latex form. •There is an option Get Latex but there is no “Paste Latex” option •When I tried to import something using option “Paste plain text” all the whitespaces/new line were lost in imported material. Please solve these issues as soon as possible

$${To}\:{Tinku}\:{Tara}\:{sir}, \\ $$$${How}\:{can}\:{I}\:{import}\:{something}\:{in}\:{Latex}\:{form}. \\ $$$$\bullet{There}\:{is}\:{an}\:{option}\:\mathrm{Get}\:\mathrm{Latex}\:{but}\:{there}\:{is}\:{no} \\ $$$$``\mathrm{Paste}\:\mathrm{Latex}''\:{option} \\ $$$$\bullet{When}\:{I}\:{tried}\:{to}\:{import}\:{something}\:{using}\:{option} \\ $$$$``\mathrm{Paste}\:\mathrm{plain}\:\mathrm{text}''\:{all}\:{the}\:{whitespaces}/{new}\:{line}\: \\ $$$${were}\:{lost}\:{in}\:{imported}\:{material}. \\ $$$$\boldsymbol{{Please}}\:{solve}\:{these}\:{issues}\:{as}\:{soon}\:{as}\:{possible} \\ $$

Question Number 217071 Answers: 1 Comments: 0

Find all two-digit numbers that are equal to four times the sum of their digits. Solve this using at least two different methods and verify your answers.

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{two}-\mathrm{digit}\:\mathrm{numbers}\:\mathrm{that}\:\mathrm{are}\:\mathrm{equal}\:\mathrm{to}\:\mathrm{four}\:\mathrm{times}\:\mathrm{the}\:\mathrm{sum}\: \\ $$$$\mathrm{of}\:\mathrm{their}\:\mathrm{digits}.\:\mathrm{Solve}\:\mathrm{this}\:\mathrm{using}\:\mathrm{at}\:\mathrm{least}\:\mathrm{two}\:\mathrm{different}\:\mathrm{methods}\: \\ $$$$\mathrm{and}\:\mathrm{verify}\:\mathrm{your}\:\mathrm{answers}. \\ $$

Question Number 217064 Answers: 1 Comments: 0

Two numbers differ by 6. The sum of their reciprocals is (2/(15)) . Determine the numbers.

$$\mathrm{Two}\:\mathrm{numbers}\:\mathrm{differ}\:\mathrm{by}\:\mathrm{6}.\:\mathrm{The}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{their}\:\mathrm{reciprocals}\:\mathrm{is}\:\frac{\mathrm{2}}{\mathrm{15}}\:. \\ $$$$\mathcal{D}{etermine}\:{the}\:{numbers}. \\ $$

Question Number 217079 Answers: 1 Comments: 0

((6C3×4C1)/(15C4))

$$\frac{\mathrm{6}{C}\mathrm{3}×\mathrm{4}{C}\mathrm{1}}{\mathrm{15}{C}\mathrm{4}} \\ $$$$ \\ $$

Question Number 217050 Answers: 1 Comments: 1

Question Number 217046 Answers: 1 Comments: 0

form the differential equation by eliminating the arbritrary constant y^2 =Ax^2 +Bx+C

$${form}\:{the}\:{differential}\:{equation}\:{by}\: \\ $$$${eliminating}\:{the}\:{arbritrary}\:{constant} \\ $$$${y}^{\mathrm{2}} ={Ax}^{\mathrm{2}} +{Bx}+{C} \\ $$

Question Number 217049 Answers: 0 Comments: 1

Hello here its been a while. I′ve been watching Ajfour sir dancing lately on Youtube. I believe he has gotten to saturation point. Is there anyone here into AI and Machine learning?

$${Hello}\:{here}\:{its}\:{been}\:{a}\:{while}.\:{I}'{ve} \\ $$$${been}\:{watching}\:{Ajfour}\:{sir}\:{dancing} \\ $$$${lately}\:{on}\:{Youtube}.\:{I}\:{believe}\:{he}\:{has} \\ $$$${gotten}\:{to}\:{saturation}\:{point}. \\ $$$${Is}\:{there}\:{anyone}\:{here}\:{into}\:{AI}\:{and} \\ $$$${Machine}\:{learning}? \\ $$$$ \\ $$

Question Number 217040 Answers: 2 Comments: 0

Find all positive integers n such that n divides 2^n + 1.

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{positive}\:\mathrm{integers}\:\:\mathrm{n}\:\:\mathrm{such}\:\mathrm{that} \\ $$$$\:\:\mathrm{n}\:\:\mathrm{divides}\:\:\mathrm{2}^{{n}} \:+\:\mathrm{1}.\:\: \\ $$

Question Number 217058 Answers: 1 Comments: 0

The quadratic equation has two equal roots: x^2 +(k−3)x+k=0 (a) Find the value of k. (b) For this value of k, solve the equation for x (c)If x is the length of a rectangle and its width is x−2, find the area of the rectangle.

$$\mathrm{The}\:\mathrm{quadratic}\:\mathrm{equation}\:\:\mathrm{has}\:\mathrm{two}\:\mathrm{equal}\:\mathrm{roots}: \\ $$$$\:\:\:\:\:{x}^{\mathrm{2}} +\left({k}−\mathrm{3}\right){x}+{k}=\mathrm{0} \\ $$$$\left(\mathrm{a}\right)\:\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\:{k}. \\ $$$$\left(\mathrm{b}\right)\:\mathrm{For}\:\mathrm{this}\:\mathrm{value}\:\mathrm{of}\:\:{k},\:\mathrm{solve}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{for}\:{x} \\ $$$$\left(\mathrm{c}\right)\mathrm{If}\:\:{x}\:\mathrm{is}\:\mathrm{the}\:\mathrm{length}\:\mathrm{of}\:\mathrm{a}\:\mathrm{rectangle}\:\mathrm{and}\:\mathrm{its}\:\mathrm{width}\:\mathrm{is}\:{x}−\mathrm{2},\:\:\mathrm{find}\:\mathrm{the}\:\mathrm{area}\: \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{rectangle}. \\ $$

Question Number 217032 Answers: 0 Comments: 0

Question Number 217024 Answers: 1 Comments: 0

Question Number 217030 Answers: 1 Comments: 0

Find all positive integers n such that n + 1 divides n^2 + 1

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{positive}\:\mathrm{integers}\:\:\mathrm{n}\:\:\mathrm{such}\:\mathrm{that}\:\: \\ $$$$\:\mathrm{n}\:+\:\mathrm{1}\:\:\mathrm{divides}\:\:\mathrm{n}^{\mathrm{2}} \:+\:\mathrm{1} \\ $$

Question Number 216998 Answers: 0 Comments: 3

Question Number 216995 Answers: 3 Comments: 0

Find all prime numbers p and q such that p^2 − q^2 = 2024

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{prime}\:\mathrm{numbers}\:\:\mathrm{p}\:\mathrm{and}\:\mathrm{q}\: \\ $$$$\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{p}^{\mathrm{2}} −\:\:\mathrm{q}^{\mathrm{2}} =\:\:\mathrm{2024} \\ $$

Question Number 217015 Answers: 2 Comments: 0

If f(2x + 1) = 3x + 5 Find f(x) = ?

$$\mathrm{If} \\ $$$$\mathrm{f}\left(\mathrm{2x}\:+\:\mathrm{1}\right)\:=\:\mathrm{3x}\:+\:\mathrm{5} \\ $$$$\mathrm{Find} \\ $$$$\mathrm{f}\left(\mathrm{x}\right)\:=\:? \\ $$

Question Number 216990 Answers: 2 Comments: 0

(1) ∫(sec^2 x∙(√(tan x)))dx=?

$$\left(\mathrm{1}\right)\:\int\left(\mathrm{sec}^{\mathrm{2}} {x}\centerdot\sqrt{\mathrm{tan}\:{x}}\right){dx}=? \\ $$

Question Number 216983 Answers: 1 Comments: 0

Question Number 217075 Answers: 0 Comments: 3

Help me please... $\vv{AB}$ and $\vv{CD}$ are two vectors, and $\vv{AB}$ is not the zero vector. Prove that if the vectors $\vv{AB}$ and $\vv{CD}$ are colinear, then there exists a real number $ k $ such that $ \vv{CD} = k \vv{AB} $. (don't use coordinates !)

$$ \\ $$Help me please... $\vv{AB}$ and $\vv{CD}$ are two vectors, and $\vv{AB}$ is not the zero vector. Prove that if the vectors $\vv{AB}$ and $\vv{CD}$ are colinear, then there exists a real number $ k $ such that $ \vv{CD} = k \vv{AB} $. (don't use coordinates !)

Question Number 216959 Answers: 0 Comments: 0

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