Let N be the greatest multiple of 36 all of whose digits are even and no two of whose digits are the same. Find the remainder when N is divided by 1000.


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Question Number 111831 by Aina Samuel Temidayo last updated on 05/Sep/20

$Let N be the greatest multiple of 36 all$ $of whose digits are even and no two of$ $whose digits are the same . Find the$ $remainder when N is divided by 1000 .$
	Answered by Rasheed.Sindhi last updated on 05/Sep/20

	$36 = 4 \times 9$ $^{▸} A n u m b e r i s d i v i s i b l e b y 36 i f i t$ $i s d i v i s i b l e b y 4 & 9$ $^{▸} A n u m b e r i s d i v i s i b l e b y 9 i f$ $s u m o f d i g i t s o f t h e n u m b e r i s$ $d i v i s i b l e b y 9 .$ $^{▸} A n u m b e r i s d i v i s i b l e b y 4 i f$ $t h e t w o d i g i t n u m b e r c o m p o s e d$ $o f {o n e}^{'} s & t e n s i s m u l t i p l e o f 4$ $N c o n s i s t s o f d i s t i n c t & e v e n$ $d i g i t s : 0, 2, 4, 6, 8$ $O n l y p o s s i b l e c o m b i n a t i o n o f d i g i t s f o r$ $w h i c h t h e f o r m e d n u m b e r m a y b e$ $d i v i s i b l e b y 9 i s 0, 4, 6, 8 . (s u m$ $w i l l b e 0 + 4 + 6 + 8 = 18 w h i c h i s$ $d i v i s i b l e b y 9 .$ $N i s g r e a t e s t s o m o s t s i g n i f i c a n t$ $d i g i t m u s t b e 8 a n d n e x t b e 6 i f$ $p o s s i b l e .$ $8 6__.$ $T h e r e m a i n i n g d i g i t s a r e 0 & 4$ $T w o 2 - d i g i t n u m b e r s a r e p o s s i b l e$ $04 & 40 . B o t h n u m b e r s a r e$ $f o r t u n a t e l y d i v i s i b l e b y 4 . B u t$ $f o r N (t h e g r e a t e s t) 40 i s o n l y$ $o p t i o n .$ $^{∙} S o n o w t h e n u m b e r i s 8640 .$ $^{∙} T h e r e m a i n d e r o n d i v i d i n g b y$ $1000 i s o b v i o u s l y :$ $640$
	Commented by Aina Samuel Temidayo last updated on 05/Sep/20

	$Cool . Thanks .$
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