The smallest integer numbers with n ≥ 2018 so ((√3)+3i)^n form real numbers is..


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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

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