how-many-6-digit-numbers-exist-which-are-divisible-by-11-and-have-no-repeating-digits-

Question Number 115408 by mr W last updated on 25/Sep/20

h o w m a n y 6 d i g i t n u m b e r s e x i s t

w h i c h a r e d i v i s i b l e b y 11 a n d h a v e n o

r e p e a t i n g d i g i t s ?

Answered by Olaf last updated on 26/Sep/20

N = a_{5} a_{4} a_{3} a_{2} a_{1} a_{0}

A number is divisible by 11 if the sum

of its even - numbered digits

substracted from the sum of its

odd - numbered digits is zero

or a multiple of 11 .

N = a_{0} + 10 a_{1} + 10^{2} a_{2} + 10^{3} a_{3} + 10^{4} a_{4} + 10^{5} a_{5}

N = a_{0} + (1 \times 11 - 1) a_{1} + (9 \times 11 + 1) a_{2}

+ (91 \times 11 - 1) a_{3} + (909 \times 11 + 1) a_{4} + (9091 \times 11 - 1) a_{5}

\Rightarrow N = (a_{0} - a_{1} + a_{2} - a_{3} + a_{4} - a_{5}) + 11 p

with p \in Z

\Rightarrow N is divisible by 11 if

a_{0} - a_{1} + a_{2} - a_{3} + a_{4} - a_{5} \equiv 0 [11]

(a_{0} + a_{2} + a_{4}) - (a_{1} + a_{3} + a_{5}) \equiv 0 [11]

\dots to be continued .

Commented by mr W last updated on 26/Sep/20

t h a n k s s o f a r s i r!

s e e m s t o b e a t o u g h t a s k, s i n c e w e

h a v e C_{6}^{10} = 210 w a y s t o s e l e c t 6 d i g i t s!

Facebook Tweet Pin