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-

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 .