z-is-a-complex-number-with-Re-z-Im-z-N-Determine-z-if-z-z-1000-

Question Number 111159 by Rasheed.Sindhi last updated on 02/Sep/20

z i s a c o m p l e x n u m b e r w i t h

R e (z), I m (z) \in N .

D e t e r m i n e z i f

z . \bar{z} = 1000

Answered by Sarah85 last updated on 02/Sep/20

{(R e (z))}^{2} + {(I m (z))}^{2} = 1000

\Rightarrow

z = 10 + 30 i \lor 18 + 26 i \lor 26 + 18 i \lor 30 + 10 i

Commented by Rasheed.Sindhi last updated on 02/Sep/20

TH α n X m i s s! A n y p r o c e s s ?

Commented by Sarah85 last updated on 02/Sep/20

a = \sqrt{1000 - b^{2}} a n d 1 ⩽ b ⩽ 31

Commented by Rasheed.Sindhi last updated on 03/Sep/20

O t h e r m e t h o d w h i c h m a k e s

t h e s e a r c h m o r e n a r r o w .

Commented by Rasheed.Sindhi last updated on 03/Sep/20

You can't use 'macro parameter character #' in math mode

Answered by Rasheed.Sindhi last updated on 04/Sep/20

z = a + i b

z . \bar{z} = 1000 \Rightarrow a^{2} + b^{2} = 1000

1000 i s d o u b l y e v e n n u m b e r,

i - e {i t}^{'} s d i v i s i b l e b y 4 . T h i s i m p l i e s

t h a t a & b a r e b o t h e v e n :

1000 \in E \Rightarrow a^{2} + b^{2} \in E

\Rightarrow (a^{2}, b^{2} \in E) \lor (a^{2}, b^{2} \in O)

\Rightarrow (a, b \in E) \lor (a, b \in O)

B u t (a^{2} + b^{2}) i s d o u b l y e v e n

s o a, b \notin O : l e t a = 2 p + 1, b = 2 q + 1

a^{2} + b^{2} = {(2 p + 1)}^{2} + {(2 q + 1)}^{2}

= 4 (p^{2} + q^{2} + p + q) + 2 (s i n g l y e v e n)

∴ a, b \in E

T h i s n a r r o w s t h e s e a r c h . O n l y

w e h a v e t o l o o k f o r 2, 4, 6, \dots, 30 n o w

T o m a k e t h e s e a r c h m o r e n a r r o w :

L e t a = 10 t_{a} + u_{a}, b = 10 t_{b} + u_{b}

a^{2} + b^{2} = {(10 t_{a} + u_{a})}^{2} + {(10 t_{b} + u_{b})}^{2}

(100 t_{a}^{2} + 20 t_{a} u_{a} + u_{a}^{2}) + (100 t_{b}^{2} + 20 t_{b} u_{b} + u_{b}^{2}) = 1000

\Rightarrow U n i t - d i g i t o f u_{a}^{2} + u_{b}^{2}

= U n i t - d i g i t o f 1000 = 0

O n l y p o s s i b i l i t i e s a r e :

0^{2} + 0^{2} = 0

1^{2} + 3^{2} = 10 (e x c l u d e d d u e t o 1, 3 \in O),

2^{2} + 6^{2} = 40

3^{2} + 9^{2} = 90 (e x c l u d e d d u e t o 1, 3 \in O)

2^{2} + 4^{2} = 20

5^{2} + 5^{2} = 50 (e x c l u d e d d u e t o 1, 3 \in O)

6^{2} + 8^{2} = 100

p o s s i b l e u n i t s : 0, 2, 4, 6, 8

N o h e l p f r o m t h i s s e c o n d p a r t

w e h a v e t o l o o k f o r a l l e v e n

n u m b e r s u p t o 30 .

S o f i n a l l y,

z = 10 + 30 i \lor 30 + 10 i \lor 18 + 26 i \lor 26 + 18 i