Given: N in base 10 is 158b687a where b and a are digits (a<b). Given: N≡2+a[4]. 1) Determinate couples (a;b) such that 11 divise N. 2) Determinate couples (a;b) such that 3 and 25 divise N. thanks.


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Question Number 121568 by mathocean1 last updated on 09/Nov/20

$Given : N in base 10 is$ $158 b 687 a where b and a are$ $digits (a < b) . Given :$ $N \equiv 2 + a [4] .$ $1) Determinate couples$ $(a; b) such that 11 divise N .$ $2) Determinate couples (a; b)$ $such that 3 and 25 divise N .$ $thanks .$
	Answered by kolos last updated on 17/Nov/20

	$1) j f 11 d i v i s e N, t h e n N \equiv 0 m o d (11)$ $15806870 + 10000 b + a \equiv 0 m o d (11)$ $2 + b + a \equiv o m o d (11)$ $a + b \equiv 9 m o d 11$ $(a, b) = (0, 9) (1, 8) (2, 7) . . . (8, 1) (9, 0)$ $2) i f N i s m u l t i p l e o f 25, t h e r e f o r e$ $t h e l a s t t w o d i g i t s a r e 00, 25, 50 o r 75$ $a s t h e l a s t t w o d i g i t s o f N a r e 7 a, t h e n$ $a = 5$ $t o b e m u l t i p l e o f 3, t h e s u m o f t h e$ $d i g i t s a r e a m u l t i p l e t o o$ $t h e n$ $1 + 5 + 8 + b + 6 + 8 + 7 + 5 \equiv 0 m o d 3$ $b + 40 \equiv 0 m o d 3$ $b \equiv 2 m o d 3$ $b = 2, 5, 8$ $(a, b) = (5, 2) (5, 5) (5, 8)$ $I h o p e i^{'} v e h e l p e d :)$
	Commented bymathocean1 last updated on 19/Nov/20

	$t h a n k y o u v e r y m u c h s i r!$
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