The-smallest-integer-numbers-with-n-2018-so-3-3i-n-form-real-numbers-is-

Question Number 55674 by gunawan last updated on 01/Mar/19

The smallest integer numbers

with n ⩾ 2018 so

{(\sqrt{3} + 3 i)}^{n} form real numbers is . .

Answered by mr W last updated on 02/Mar/19

{(\sqrt{3} + 3 i)}^{n}

= {(2 \sqrt{3})}^{n} {(\frac{1}{2} + \frac{\sqrt{3}}{2} i)}^{n}

= {(2 \sqrt{3})}^{n} {(\cos \frac{π}{3} + i \sin \frac{π}{3})}^{n}

= {(2 \sqrt{3})}^{n} (\cos \frac{n π}{3} + i \sin \frac{n π}{3})

\Rightarrow \sin \frac{n π}{3} = 0

\Rightarrow \frac{n π}{3} = m π

\Rightarrow n = 3 m ⩾ 2018

\Rightarrow n_{m i n} = 2019

Commented by gunawan last updated on 02/Mar/19

Nice thank you Sir

Commented by malwaan last updated on 02/Mar/19

what is the point ?

we can write 2019

without proof

please explane sir

Commented by mr W last updated on 02/Mar/19

\Rightarrow n = 3 m ⩾ 2018

m e a n s n m u s t b e d i v i s i b l e b y 3 a n d

a t l e a s t 2018 .

a n u m b e r i s d i v i s i b l e b y 3 i f t h e

s u m o f i t s d i g i t s i s d i v i s i b l e b y 3 .

w i t h 2018 t h e s u m o f i t s d i g i t s i s

2 + 0 + 1 + 8 = 11, n o t d i v i s i b l e b y 3,

b u t 2 + 0 + 1 + 9 = 12, {i t}^{'} s d i v i s i b l e b y 3,

t h e r e f o r e n_{m i n} = 2019 .

Commented by malwaan last updated on 03/Mar/19

thank you very much sir

Commented by Kunal12588 last updated on 03/Mar/19

s i r i h e a r d t h i s

f o r c h e c k i n g d i v i s i b i l i t y w i t h 3

w e h a v e t o a d d t h e d i g i t s o f t h e n u m b e r

s p e c i a l c a s e : - i f a d i g i t c o n t a i n s 9

w e c a n d r o p t h e 9 a n d a d d o t h e r d i g i t s

e g : - 2019 - 2 + 0 + 1 = 3 w h i c h i s d i v i s i b l e b y 3 .

w h y i s t h i s s i r ?

Commented by mr W last updated on 03/Mar/19

b e c a u s e d i g i t 9 i s a l r e a d y d i v i s i b l y b y 3 .

i f t h e s u m o f o t h e r d i g i t s i s d i v i s i b l e

b y 3, t h e n t h e t o t a l s u m o f a l l d i g i t s i s

a l s o d i v i s i b l e b y 3 a n d t h e n u m b e r i s

d i v i s i b l e b y 3 . i n f a c t y o u c a n d r o p n o t

o n l y d i g i t 9, b u t a l s o d i g i t s 6, 3 a n d 0,

a n d o n l y n e e d t o a d d o t h e r r e m a i n i n g

d i g i t s .

e . g . w e t a k e t h e n u m b e r

123450906497628005

w e o n l y n e e d t o a d d t h e b l u e d i g i t s :

1 + 2 + 4 + 5 + 4 + 7 + 2 + 8 + 5 = 38 w h i c h

i s n o t d i v i s i b l e b y 3, t h e r e f o r e t h e

g i v e n n u m b e r i s n o t d i v i s i b l e b y 3 .

Commented by Kunal12588 last updated on 03/Mar/19

g r e a t s i r . a n d t h a n k s