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Number TheoryQuestion and Answers: Page 9

Question Number 133757 Answers: 1 Comments: 0

Question Number 133611 Answers: 1 Comments: 0

Given system of equation { ((2x−3y = 13)),((3x+2y = b)) :} , where l ≤ b≤ 100 and b is integer. Suppose n^2 = x+y where x,y is solution of given system of equation , find the value of n for n is integer

$Given system of equation$ ${\begin{cases} 2 x - 3 y = 13 \\ 3 x + 2 y = b \end{cases}, where l ⩽ b ⩽ 100 and$ $b is integer . Suppose n^{2} = x + y where$ $x, y is solution of given system$ $of equation, find the value of n$ $for n is integer$

Question Number 133412 Answers: 1 Comments: 0

What are the last two digits of 2^(222) −1 ?

$What are the last two digits of 2^{222} - 1 ?$

Question Number 133320 Answers: 1 Comments: 0

Find all n for which n^2 +2n+4 is divisible by 7

$Find all n for which n^{2} + 2 n + 4$ $is divisible by 7$

Question Number 132386 Answers: 1 Comments: 0

Find the value of { ((ln i)),((ln (3+4i))) :}

$Find the value of {\begin{cases} \ln i \\ \ln (3 + 4 i) \end{cases}$

Question Number 131497 Answers: 0 Comments: 1

please recommend problem and exercise book for number theory where answers and solutions are not given or only given at the end of the book. i find it annoying when answers are always right next to the question. a book that has no solutions would be even greater. the only such book i have found is 250 problems in elementary number theory. thank.

$please recommend problem and exercise book for number theory$ $where answers and solutions are not given or only$ $given at the end of the book . i find it annoying when answers are always right next to the question .$ $a book that has no solutions would be even greater .$ $the only such book i have found is 250 problems in elementary number theory .$ $thank .$

Question Number 131484 Answers: 3 Comments: 0

Express Σ_(n=1) ^(49) (1/( (√(n+(√(n^2 −1)))))) as a+b(√2) for some integers a and b

$Express \underset{n = 1}{\sum^{49}} \frac{1}{\sqrt{n + \sqrt{n^{2} - 1}}} as a + b \sqrt{2}$ $for some integers a and b$

Question Number 131196 Answers: 1 Comments: 0

The minimum value of the expression B = ∣z∣^2 +∣z−3∣^2 +∣z−6i∣^2 is p. What the value of (p/(10)) .?

$T h e m i n i m u m v a l u e o f t h e$ $e x p r e s s i o n B = ∣ z ∣^{2} + ∣ z - 3 ∣^{2} + ∣ z - 6 i ∣^{2}$ $i s p . W h a t t h e v a l u e o f \frac{p}{10} . ?$

Question Number 130727 Answers: 2 Comments: 2

Question Number 130109 Answers: 3 Comments: 0

2+(3/2^3 )+(4/3^3 )+(5/4^3 )+(6/5^3 )+∙∙∙∙∙∙∙∙∙=?

$2 + \frac{3}{2^{3}} + \frac{4}{3^{3}} + \frac{5}{4^{3}} + \frac{6}{5^{3}} + \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot = ?$

Question Number 128740 Answers: 1 Comments: 0

Σ_(n = 0) ^∞ (1/((n+1)(n+3)n!)) =?

$\underset{n = 0}{\sum^{\infty}} \frac{1}{(n + 1) (n + 3) n!} = ?$

Question Number 127979 Answers: 2 Comments: 0

(1+i)^(2020) =?

${(1 + i)}^{2020} = ?$

Question Number 127940 Answers: 2 Comments: 0

{ ((3x=1 (mod 4))),((4x=3 (mod 5) )),((5x=7 (mod 11))) :}

${\begin{cases} 3 x = 1 (\mod 4) \\ 4 x = 3 (\mod 5) \\ 5 x = 7 (\mod 11) \end{cases}$

Question Number 127610 Answers: 2 Comments: 0

Z=1+(1+i)cos𝛉 arg(z)=?

$Z = 1 + (1 + i) c o s θ$ $a r g (z) = ?$

Question Number 127111 Answers: 3 Comments: 0

find the value of x such that { ((x=2 (mod 5))),((x=3 (mod 8) )),((x=2 (mod 3))) :}

$f i n d t h e v a l u e o f x s u c h t h a t$ ${\begin{cases} x = 2 (m o d 5) \\ x = 3 (m o d 8) \\ x = 2 (m o d 3) \end{cases}$

Question Number 126682 Answers: 2 Comments: 0

Find the remainder 7+77+777+7777+...+777...7_(2020 times) when divided by 9.

$F i n d t h e r e m a i n d e r$ $7 + 77 + 777 + 7777 + . . . + \underset{2020 t i m e s}{\underset{⏟}{777 . . . 7}}$ $w h e n d i v i d e d b y 9 .$

Question Number 126677 Answers: 1 Comments: 3

Question Number 126666 Answers: 3 Comments: 0

3+33+333+3333+...+3333...3_(2020 times) divide by 2. find the remainder

$3 + 33 + 333 + 3333 + . . . + \underset{2020 t i m e s}{\underset{⏟}{3333 . . . 3}}$ $d i v i d e b y 2 . f i n d t h e r e m a i n d e r$

Question Number 126657 Answers: 2 Comments: 0

Question Number 126269 Answers: 1 Comments: 0

If n is an odd integer , show that 32 divides (n^2 +3)(n^3 +7) ?

$I f n i s a n o d d i n t e g e r, s h o w t h a t 32$ $d i v i d e s (n^{2} + 3) (n^{3} + 7) ?$

Question Number 125959 Answers: 1 Comments: 0

Question Number 125666 Answers: 0 Comments: 0

solution : x′ = (((−2 1)),((1 −2)) )x + (((2e^(−m) )),((3m)) )

$s o l u t i o n :$ $x^{'} = (\begin{matrix} - 2 1 \\ 1 - 2 \end{matrix}) x + (\begin{matrix} 2 e^{- m} \\ 3 m \end{matrix})$

Question Number 125042 Answers: 0 Comments: 3

2+2+2=6 3×3−3=6 (√4)+(√4)+(√4)=6 5 + 5 ÷ 5 =6 6+6−6=6 7 − 7 ÷7=6 8 8 8=6 (√9)×(√9) − (√9)=6 complete for 8

$2 + 2 + 2 = 6$ $3 \times 3 - 3 = 6$ $\sqrt{4} + \sqrt{4} + \sqrt{4} = 6$ $5 + 5 \div 5 = 6$ $6 + 6 - 6 = 6$ $7 - 7 \div 7 = 6$ $8 8 8 = 6$ $\sqrt{9} \times \sqrt{9} - \sqrt{9} = 6$ $c o m p l e t e f o r 8$

Question Number 124944 Answers: 0 Comments: 0

1. (a, m)=(b, m)=1⇒(ab, m)=1 2. c∣ab and (c, a)=1⇒c∣b 3. If c is a common multiple of a and b then [a, b]∣c 4. [ma, mb]=m[a, b] for all int m>0 5. [a, b](a, b)=∣ab∣ 6. Let g>0, s be integers. Show that g∣s iff ∃ integers x, y such that s=x+y and (x, y)=g

$1 . (a, m) = (b, m) = 1 \Rightarrow (a b, m) = 1$ $2 . c ∣ a b and (c, a) = 1 \Rightarrow c ∣ b$ $3 . If c is a common multiple of$ $a and b then [a, b] ∣ c$ $4 . [m a, m b] = m [a, b] for all int m > 0$ $5 . [a, b] (a, b) =∣ a b ∣$ $6 . Let g > 0, s be integers . Show$ $that g ∣ s iff \exists integers x, y such$ $that s = x + y and (x, y) = g$

Question Number 124795 Answers: 1 Comments: 0

Question Number 124130 Answers: 1 Comments: 0

Let P = ( 3(√6) + 7 )^(89) and F is fractional part of P. Then find the remainder when (PF) + (PF)^2 + (PF)^3 is divided by 31.

$Let P = {(3 \sqrt{6} + 7)}^{89} and F is fractional$ $part of P .$ $Then find the remainder when$ $(PF) + {(PF)}^{2} + {(PF)}^{3} is divided by 31 .$

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