How many 3 digited numbers which are divisible by 1)3 2)4 3)5 4)6 5)7 6)8 7)9 with repetetion of digits is NOT allowed...one problem process


Question and Answers Forum

All Questions Topic List
Permutation and Combination Questions
Previous in All Question Next in All Question
Previous in Permutation and Combination Next in Permutation and Combination
Question Number 177309 by SLVR last updated on 03/Oct/22

$H o w m a n y 3 d i g i t e d n u m b e r s$ $w h i c h a r e d i v i s i b l e b y 1) 3$ $2) 4 3) 5 4) 6 5) 7 6) 8 7) 9$ $w i t h r e p e t e t i o n o f d i g i t s i s$ $N O T a l l o w e d . . . o n e p r o b l e m p r o c e s s$
Commented by SLVR last updated on 03/Oct/22

$i n e e d f o r d i v i s i b l e b y 7$
	Answered by mahdipoor last updated on 04/Oct/22

	$N = (a b c) = 100 a + 10 b + c$ $0 ⩽ a, b, c \in N ⩽ 9 a \neq 0$ $\Rightarrow N \overset{7}{\equiv} 0 a n d r e p e t e t i o n o f d i g i t s$ $\Rightarrow 100 a + 10 b + c \overset{7}{\equiv} 2 a + 3 b + c \overset{7}{\equiv} 0$ $i) a = b \Rightarrow c \overset{7}{\equiv} 2 a \Rightarrow 13 n u m b e r s$ $i i) a = c \Rightarrow b \overset{7}{\equiv} 6 a \Rightarrow 12$ $i i i) b = c \Rightarrow a \overset{7}{\equiv} 5 b \Rightarrow 12$ $i v) a = b = c \Rightarrow 6 a \overset{7}{\equiv} 0 \Rightarrow 1$ $\Rightarrow a l l n u m b e r s = 13 + 12 + 12 - 2 \times 1 = 35$ $3 d i g i t n u m b e r a r e d i v i s i b l e b y 7 :$ $[\frac{999}{7}] - [\frac{99}{7}] = 128$ $\Rightarrow 128 - 35 = 93$
	Commented by SLVR last updated on 04/Oct/22

	$T h a n k s f o r y o u r c o n c e r n s i r$ $b u t h o w 4 t h l i n e 2 a + 3 b + c \equiv 0 m o d 7$ $a n d h o w 13 n u m b e r s i f a = b$
	Commented by SLVR last updated on 04/Oct/22

	$k i n d l y e x p l a i n w h a t t h o s e 34$ $n u m b e r s b e d e d u c t e d . . p l e a s e$
	Commented by mr W last updated on 04/Oct/22

	$i i) a = c$ $2 a + 3 b + c = 3 (a + b) \overset{7}{\equiv} 0$ $a + b = 7 \Rightarrow 7$ $a + b = 14 \Rightarrow 5$ $t o t a l 12 n o t 11 ?$ $i c o u n t e d t o t a l l y 93 v a l i d n u m b e r s .$
	Commented by mr W last updated on 04/Oct/22

	Commented by mahdipoor last updated on 04/Oct/22

	$a n s w e r t o S L V R :$ $i >$ $100 a + 10 b + c \overset{7}{\equiv} (98 a + 7 b) + (2 a + 3 b + c) \overset{7}{\equiv}$ $2 a + 3 b + c 7 ∣ 98 a + 7 b = 7 (14 a + b)$ $i i > f o r e x a m p l e :$ $a = c \Rightarrow 3 (a + b) \overset{7}{\equiv} (a + b) \overset{7}{\equiv} 0 \Rightarrow$ $b \overset{7}{\equiv} - a + 7 a \Rightarrow b \overset{7}{\equiv} 6 a 0 ⩽ a, b \in N ⩽ 9 a \neq 0$ $(a, b) = (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (5, 9),$ $(6, 1), (6, 8), (7, 0), (7, 7), (8, 6), (9, 5)$ $\Rightarrow 12 n u m b e r$ $a n s w e r t o M r . W :$ $t h x, i e d i t e d .$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum