Let S= {1, 2, ..., 9, 0} A: How many multiples of eight of four digits can be formed from S B: The same question for different digits


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Question Number 178338 by Acem last updated on 15/Oct/22

$L e t S = {1, 2, . . ., 9, 0} A : H o w m a n y m u l t i p l e s$ $o f e i g h t o f f o u r d i g i t s c a n b e f o r m e d f r o m S$ $B : T h e s a m e q u e s t i o n f o r d i f f e r e n t d i g i t s$
Commented by Acem last updated on 17/Oct/22

$H i n t :$ $A n s w e r o f B i s 560 n u m b e r s w i t h d i f f e r e n t d i g i t s$
	Answered by mr W last updated on 16/Oct/22

	$[A]$ $⌊ \frac{9999}{8} ⌋ - ⌊ \frac{999}{8} ⌋ = 1249 - 124 = 1125$ $s o l u t i o n f o r A i s s i m p l e . w e j u s t n e e d$ $t o c o u n t t h e n u m b e r o f t e r m s o f a n$ $A . P . :$ $1000, 1008, 1016, . . . ., 9992$ ${t h a t}^{'} s \frac{9992 - 1000}{8} + 1 = 1125 n u m b e r s .$
	Commented by mr W last updated on 16/Oct/22

	$m a y b e t h e r e a r e b e t t e r w a y s .$
	Commented by mr W last updated on 16/Oct/22

	$[B]$ $s a y s u c h a n u m b e r i s A B C D w i t h$ $A \neq B \neq C \neq D, A \neq 0$ $\begin{array}{cccc} C D = & 08 & 16 & 24 & 32 & 40 & 48 & 56 & 64 & 72 & 80 & 96 \\ B_{n} & 3 & 3 / 1 & 2 / 1 & 3 / 1 & 3 & 2 / 1 & 4 & 2 / 1 & 3 / 1 & 3 & 3 / 1 \\ A_{n} & 7 & 6 / 7 & 6 / 7 & 6 / 7 & 7 & 6 / 7 & 6 / 7 & 6 / 7 & 6 / 7 & 7 & 6 / 7 \\ 21 & 25 & 19 & 25 & 21 & 19 & 25 & 19 & 25 & 21 & 25 \end{array}$ $21 \times 3 + 25 \times 5 + 19 \times 3 = 245$ $\begin{array}{cccc} C D = & 04 & 12 & 20 & 28 & 36 & 52 & 60 & 68 & 76 & 84 & 92 \\ B_{n} & 5 & 4 & 5 & 5 & 4 & 4 & 5 & 5 & 4 & 5 & 4 \\ A_{n} & 7 & 6 & 7 & 6 & 6 & 6 & 7 & 6 & 6 & 6 & 6 \\ 35 & 24 & 35 & 30 & 24 & 24 & 35 & 30 & 24 & 30 & 24 \end{array}$ $35 \times 3 + 24 \times 5 + 30 \times 3 = 315$ $\Rightarrow t o t a l l y 245 + 315 = 560 n u m b e r s w i t h$ $d i f f e r e n t d i g i t s$ $================$ $e x p l a n a t i o n o f t h e t a b l e$ $s a y t h e n u m b e r i s A B C D .$ $w h e n C D = 08, B c a n b e 2, 4, 6, i . e .$ $3 p o s s i b i l i t i e s (B_{n} = 3) . f o r e a c h B = 2,$ $A c a n b e 1, 3, 4, 5, 6, 7, 9, i . e . 7 p o s s i b i l i t i e s$ $(A_{n} = 7) . \Rightarrow 3 \times 7 = 21 n u m b e r s .$
	Commented by Acem last updated on 16/Oct/22

	$I h a v e a n o t h e r w a y f o r B, i h o p e {i t}^{'} s s i m p l e b u t$ $o u r r e s u l t s a r e d i f f e r e n t$
	Commented by Acem last updated on 16/Oct/22

	$M y a n s w e r f o r B i s 541,$ $T h e r e a r e 19 d i f f . n u m . b e t w e e n u s,$ $T h e s o l u t i o n i s b e l o w$
	Commented by mr W last updated on 16/Oct/22

	$p l e a s e r e c h e c k y o u r s o l u t i o n s i r .$ $t h e c o r r e c t a n s w e r i s 560 .$
	Commented by Acem last updated on 17/Oct/22

	$Y e s, b e c a u s e o n e o f t h e e x c l u d e d n u m b e r s$ $i s u b t r a c t i t t w i c e, o o o p s . A n y w a y m y m e t h o d$ $i s l e s s d i f f i c u l t o f y o u r s . I t r i e d y o u r s a n d i t$ $n e e d f o r e a c h C D m u s t w r i t e o r f i g u r e$ $2 s u b g r o u p s f o r A B, w h i l e m i n e n e e d o n l y$ $s i m p l y c o m p e n s a t i o n v a l u e s o d d & e v e n .$ $G o o d l u c k!$
	Commented by mr W last updated on 18/Oct/22

	$i n e v e r p r a i s e m y s e l f .$
	Commented by Acem last updated on 18/Oct/22

	$Y o u d o i n d i r e c t l y . B u t m e n e v e r .$ $A n y w a y i s a i d t h a t m y m e t h o d i s e a s i e r n o t t o$ $p r a i s e m y s e l f ‘ ‘ {d i d n}^{'} t t a l k m y s k i n {c o l o r}^{″}, b u t$ $f o r a l l o f u s t o u s e f u l l f r o m e a c h o t h e r,$ $a n d t a k e f r o m a l l t h e b e t t e r i d e a, e s p e c i a l l y t h e$ $e a s i e r, w h a t e v e r w h o s e!$ $B y t h e {w a y}_{1} o u r m e t h o d s a n d i d e a s a b o u t$ $t h i s q u s e t i o n a r e v e r y c l o s e$ $B y t h e {w a y}_{2} w h e n i c o m p o s e d t h i s q u e s t i o n$ $i w a s s u r e t h a t y o u w i l l l i k e i t a n d$ $s o l v e i t c o r r e c t l y .$ $S i n c e r e l y$
	Answered by Acem last updated on 19/Oct/22

	$A :$ $L a s t t w o d i g i t s$ $\begin{array}{cccccc} T_{t h} & T h & n \\ 1 & 0^{e v} & 00 & 08 & 16 & . . . & 88 & 96 & 13 & {e v e n}_{T h} \\ 2 & 1^{o d d} & 04 & 12 & 20 & . . . & 84 & 92 & 12 & {o d d}_{T h} \\ 3 & 2^{e v} & . . . & 13 & {e v e n}_{T h} \\ ⋮ & ⋮ \\ 9 & 9^{o d d} & 04 & 12 & 20 & . . . & 84 & 92 & 12 & {o d d}_{T h} \end{array}$ $\begin{array}{cc} T h e 2 {d i g}_{T h} . h a s 5_{e v e n} & 5_{o d d} \\ 9_{T_{t h}} \times 5_{e v e n} \times 13 + 9_{T_{t h}} \times 5_{o d d} \times 12 \end{array}$ $N u m N u m b ._{d i v i s i b l e b y 8} = 9 \times 5 (13 + 12) = 1 125$ $N o t e : T h i s m e t h o d i s f o r w h o {d i d n}^{'} t g e t t h e$ $t w o m e t h o d s a b o v e . A n d {i t}^{'} s u s e f u l l f o r n e x t (B)$ $B : i s n e x t s o o n$
	Commented by Acem last updated on 16/Oct/22

	$G o o d S i r! a s y o u s a i d {i t}^{'} s e a s y, b u t i t m i g h t n o t$ $c o m e t o m i n d$
	Answered by Acem last updated on 19/Oct/22

	$B :$ $N u m N u m b ._{d i v i s . b y 8, d i f f . d i g} = N_{{e v e n}_{T h}} + N_{{o d d}_{T h}} . . . (1)$ $N_{{e v e n}_{T h}}; l a s t t w o d i g a r e : 00, 08, 16, 24, . . ., 88, 96$ $\begin{array}{ccccccccccc} e v e n \\ T_{T h} & T h \\ 1^{⌣ 5} & 0^{⌣ 9} \\ 2^{⌣ 4} & 2^{⌣ 8} \\ 3 & 4 \\ 4 & 6 \\ 5 & 8^{⌣ 8} \\ 6 \\ 7 \\ 8 \\ 9 \end{array} N_{t w o 1 s t d i g} = 5_{o d d} \times 5^{T h} + 4_{e v e n} \times 4^{T h} = 41$ $T i p s :$ $0 : m u s t s u b t r a c t 9$ $e v e n : m u s t s u b t r a c t 11_{T o f i g u r e h o w i s a i d 11}^{T r y m a k i n g a n u m . x y_08}$ $o d d : m u s t s u b t r a c t 5$ $x x : i s n o t v a l i d, m u s t s u b t r a c t 41$ $08 : 9 + 11 \Rightarrow 20, 32 \Rightarrow 16, 48 \Rightarrow 22 . . . e t c$ $\begin{array}{cc} 00 & 08 & 16 & 24 & 32 & 40 & 48 & 56 & 64 & 72 & 80 & 88 & 96 & n = 13 \\ S u m & 41 & 20 & 16 & 22 & 16 & 20 & 22 & 16 & 22 & 16 & 20 & 41 & 16 \end{array}$ $S u m = 41 \times 2 + 20 \times 3 + 16 \times 5 + 22 \times 3 = 288$ $N_{{e v e n}_{T h}} = 41 \times 13 - 288 = 245 . . . (2)$ $N_{{o d d}_{T h}}; l a s t t w o d i g a r e : 04, 12, 20, 28, . . ., 84, 92$ $\begin{array}{ccccccccccc} o d d \\ T_{T h} & T h \\ 1^{⌣ 4} & 1^{⌣ 8} \\ 2^{⌣ 5} & 3^{⌣ 8} \\ 3 & 5 \\ 4 & 7 \\ 5 & 9^{⌣ 8} \\ 6 \\ 7 \\ 8 \\ 9 \end{array} N_{t w o 1 s t d i g} = 5_{o d d} \times 4^{T h} + 4_{e v e n} \times 5^{T h} = 40$ $T i p s :$ $e v e n : m u s t s u b t r a c t 5$ $o d d : m u s t s u b t r a c t 11$ $x x : i s n o t v a l i d, m u s t s u b t r a c t 40$ $\begin{array}{cc} 04 & 12 & 20 & 28 & 36 & 44 & 52 & 60 & 68 & 76 & 84 & 92 & n = 12 \\ S u m & 5 & 16 & 5 & 10 & 16 & 40 & 16 & 5 & 10 & 16 & 10 & 16 \end{array}$ $S u m = 5 \times 3 + 16 \times 5 + 10 \times 3 + 40 \times 1 = 165$ $N_{{o d d}_{T h}} = 40 \times 12 - 165 = 315 . . . (3)$ $B y c o m p e n s a t e (2) & (3) i n (1)$ $N u m N u m b ._{d i v i s . b y 8, d i f f . d i g} = 245 + 315 = 560$
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