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{ (( ab ^(−) ∙ b ^(−) + ba ^(−) ∙ a ^(−) = cde ^(−) )),(( ab ^(−) ∙ b ^(−) − ba ^(−) ∙ a ^(−) = f ^(−) )) :} a,b,c,d,e,f are all different and in some order consecutive also. Determine the remaining decimal digits.

$$\:\begin{cases}{\overline {\:{ab}\:}\centerdot\overline {\:{b}\:}+\overline {\:{ba}\:}\centerdot\overline {\:{a}\:}=\overline {\:{cde}\:}}\\{\overline {\:{ab}\:}\centerdot\overline {\:{b}\:}−\overline {\:{ba}\:}\centerdot\overline {\:{a}\:}=\overline {\:{f}\:}\:}\end{cases} \\ $$$${a},{b},{c},{d},{e},{f}\:{are}\:{all}\:{different}\:{and}\:{in} \\ $$$${some}\:{order}\:{consecutive}\:{also}. \\ $$$$\: \\ $$$$\mathcal{D}{etermine}\:{the}\:{remaining}\:{decimal} \\ $$$${digits}. \\ $$

Question Number 200200 Answers: 1 Comments: 0

Find four positive integers, each not exceeding 70000 and each having more than 100 divisors.

$${Find}\:{four}\:{positive}\:{integers}, \\ $$$$\:{each}\:{not}\:{exceeding}\:\mathrm{70000}\:{and}\: \\ $$$${each}\:{having}\:{more}\:{than}\:\mathrm{100} \\ $$$$\:{divisors}. \\ $$

Question Number 200041 Answers: 3 Comments: 0

By strong induction prove that any natural number equal to or bigger than 8 can be written as 3a+5b where a and b are non−negative integers.

$${By}\:{strong}\:{induction}\:{prove}\:{that}\:{any} \\ $$$${natural}\:{number}\:{equal}\:{to}\:{or}\:{bigger}\:{than} \\ $$$$\mathrm{8}\:{can}\:{be}\:{written}\:{as}\:\mathrm{3}{a}+\mathrm{5}{b}\:{where}\:{a}\:{and}\:{b} \\ $$$${are}\:{non}−{negative}\:{integers}. \\ $$

Question Number 199864 Answers: 1 Comments: 0

Find the remainder Σ_(n=1) ^(2019) n^4 when divide by 53

$$\:\:\:\mathrm{Find}\:\mathrm{the}\:\mathrm{remainder}\:\underset{\mathrm{n}=\mathrm{1}} {\overset{\mathrm{2019}} {\sum}}\mathrm{n}^{\mathrm{4}} \:\mathrm{when} \\ $$$$\:\:\mathrm{divide}\:\mathrm{by}\:\mathrm{53}\: \\ $$

Question Number 199331 Answers: 0 Comments: 0

What is the remainder when 1^1 +2^2 +3^3 +......+2023^(2023) is divided by 7

$${What}\:{is}\:{the}\:{remainder}\:{when} \\ $$$$\mathrm{1}^{\mathrm{1}} +\mathrm{2}^{\mathrm{2}} +\mathrm{3}^{\mathrm{3}} +......+\mathrm{2023}^{\mathrm{2023}} \:{is}\:{divided}\:{by}\:\mathrm{7} \\ $$

Question Number 199311 Answers: 2 Comments: 0

Find the number of positive integers that are factors of 3^(19) .7^(12) .10^(25) and are also multiples of 3^(15) .7^(10) .10^(19)

$${Find}\:{the}\:{number}\:{of}\:{positive}\:{integers} \\ $$$${that}\:{are}\:{factors}\:{of}\:\mathrm{3}^{\mathrm{19}} .\mathrm{7}^{\mathrm{12}} .\mathrm{10}^{\mathrm{25}} \:{and}\:{are} \\ $$$${also}\:{multiples}\:{of}\:\mathrm{3}^{\mathrm{15}} .\mathrm{7}^{\mathrm{10}} .\mathrm{10}^{\mathrm{19}} \\ $$

Question Number 199011 Answers: 1 Comments: 1

Sum of two irrational numbers is 1 less than their product, and 8 less than their sum of squares. Find the larger of the two numbers.

$${Sum}\:{of}\:{two}\:{irrational}\:{numbers}\:{is}\:\mathrm{1} \\ $$$${less}\:{than}\:{their}\:{product},\:{and}\:\mathrm{8}\:{less}\:{than} \\ $$$${their}\:{sum}\:{of}\:{squares}.\:{Find}\:{the}\:{larger} \\ $$$${of}\:{the}\:{two}\:{numbers}. \\ $$

Question Number 198684 Answers: 3 Comments: 0

$$\:\cancel{\zeta} \\ $$

Question Number 198418 Answers: 1 Comments: 0

20^(22) −1 = ... (mod 1000)

$$\:\mathrm{20}^{\mathrm{22}} −\mathrm{1}\:=\:...\:\left(\mathrm{mod}\:\mathrm{1000}\right) \\ $$$$ \\ $$

Question Number 198400 Answers: 3 Comments: 0

20^(11) −1 = ...(mod 1000)

$$\:\:\mathrm{20}^{\mathrm{11}} −\mathrm{1}\:=\:...\left(\mathrm{mod}\:\mathrm{1000}\right) \\ $$

Question Number 197752 Answers: 1 Comments: 0

find minimum value of m such that m^(19) = 1800 (mod 2029)

$$\:\mathrm{find}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of}\:\mathrm{m} \\ $$$$\:\mathrm{such}\:\mathrm{that}\:\mathrm{m}^{\mathrm{19}} =\:\mathrm{1800}\:\left(\mathrm{mod}\:\mathrm{2029}\right) \\ $$

Question Number 197461 Answers: 1 Comments: 0

Prove that: •∫^( x) _( 0) ((lnt)/(t^2 −1))dt=∫^( (π/2)) _( 0) arctan(xtanθ)dθ • ∫^( x) _( (1/x)) ((lnt)/(t^2 −1))arctant dt=(π/8)∫^( π) _( 0) arctan((1/2)(x−(1/x))sint)dt

$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\bullet\underset{\:\mathrm{0}} {\int}^{\:\mathrm{x}} \frac{\mathrm{lnt}}{\mathrm{t}^{\mathrm{2}} −\mathrm{1}}\mathrm{dt}=\underset{\:\mathrm{0}} {\int}^{\:\frac{\pi}{\mathrm{2}}} \mathrm{arctan}\left(\mathrm{xtan}\theta\right)\mathrm{d}\theta \\ $$$$\bullet\:\:\underset{\:\frac{\mathrm{1}}{\mathrm{x}}} {\int}^{\:\mathrm{x}} \frac{\mathrm{lnt}}{\mathrm{t}^{\mathrm{2}} −\mathrm{1}}\mathrm{arctant}\:\mathrm{dt}=\frac{\pi}{\mathrm{8}}\underset{\:\mathrm{0}} {\int}^{\:\pi} \mathrm{arctan}\left(\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{x}−\frac{\mathrm{1}}{\mathrm{x}}\right)\mathrm{sint}\right)\mathrm{dt} \\ $$

Question Number 196672 Answers: 1 Comments: 1

Find all Ω=abcdef ^(−) , such that abcdef=abc+def

$${Find}\:{all}\:\Omega\overline {={abcdef}\:\:},\:{such}\:{that} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:{abcdef}={abc}+{def} \\ $$

Question Number 196676 Answers: 1 Comments: 1

If ab^(−) ∙ cd^(−) =899 , find Ω= abcd^(−) + cdab^(−)

$${If}\:\overline {{ab}}\:\centerdot\:\overline {{cd}}=\mathrm{899}\:,\:{find}\:\Omega=\:\overline {{abcd}}\:+\:\overline {{cdab}} \\ $$

Question Number 196053 Answers: 1 Comments: 0

faind n terme 4,−2,((16)/9),−2,......

$${faind}\:{n}\:{terme} \\ $$$$\mathrm{4},−\mathrm{2},\frac{\mathrm{16}}{\mathrm{9}},−\mathrm{2},...... \\ $$

Question Number 195443 Answers: 2 Comments: 0

10^(10) +10^(10^2 ) +10^(10^3 ) +...+10^(10^(10) ) ≡^7 ?