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Number TheoryQuestion and Answers: Page 3

Question Number 200315 Answers: 0 Comments: 3

{ (( 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.

${\begin{cases} \overset{―}{a b} \cdot \overset{―}{b} + \overset{―}{b a} \cdot \overset{―}{a} = \overset{―}{c d e} \\ \overset{―}{a b} \cdot \overset{―}{b} - \overset{―}{b a} \cdot \overset{―}{a} = \overset{―}{f} \end{cases}$ $a, b, c, d, e, f a r e a l l d i f f e r e n t a n d i n$ $s o m e o r d e r c o n s e c u t i v e a l s o .$ $D e t e r m i n e t h e r e m a i n i n g d e c i m a l$ $d i g i t s .$

Question Number 200200 Answers: 1 Comments: 0

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

$F i n d f o u r p o s i t i v e i n t e g e r s,$ $e a c h n o t e x c e e d i n g 70000 a n d$ $e a c h h a v i n g m o r e t h a n 100$ $d i v i s o r s .$

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.

$B y s t r o n g i n d u c t i o n p r o v e t h a t a n y$ $n a t u r a l n u m b e r e q u a l t o o r b i g g e r t h a n$ $8 c a n b e w r i t t e n a s 3 a + 5 b w h e r e a a n d b$ $a r e n o n - n e g a t i v e i n t e g e r s .$

Question Number 199864 Answers: 1 Comments: 0

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

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

Question Number 199331 Answers: 0 Comments: 0

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

$W h a t i s t h e r e m a i n d e r w h e n$ $1^{1} + 2^{2} + 3^{3} + . . . . . . + 2023^{2023} i s d i v i d e d b y 7$

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)

$F i n d t h e n u m b e r o f p o s i t i v e i n t e g e r s$ $t h a t a r e f a c t o r s o f 3^{19} . 7^{12} . 10^{25} a n d a r e$ $a l s o m u l t i p l e s o f 3^{15} . 7^{10} . 10^{19}$

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.

$S u m o f t w o i r r a t i o n a l n u m b e r s i s 1$ $l e s s t h a n t h e i r p r o d u c t, a n d 8 l e s s t h a n$ $t h e i r s u m o f s q u a r e s . F i n d t h e l a r g e r$ $o f t h e t w o n u m b e r s .$

Question Number 198684 Answers: 3 Comments: 0

$ζ$

Question Number 198418 Answers: 1 Comments: 0

20^(22) −1 = ... (mod 1000)

$20^{22} - 1 = . . . (\mod 1000)$

Question Number 198400 Answers: 3 Comments: 0

20^(11) −1 = ...(mod 1000)

$20^{11} - 1 = . . . (\mod 1000)$

Question Number 197752 Answers: 1 Comments: 0

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

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

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

$Prove that :$ $∙ {\int_{0}}^{x} \frac{lnt}{t^{2} - 1} dt = {\int_{0}}^{\frac{π}{2}} \arctan (xtan θ) d θ$ $∙ {\int_{\frac{1}{x}}}^{x} \frac{lnt}{t^{2} - 1} arctant dt = \frac{π}{8} {\int_{0}}^{π} \arctan (\frac{1}{2} (x - \frac{1}{x}) sint) dt$