what is the remainder when 767^(1009) is divided by 25


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Question Number 34140 by NECx last updated on 01/May/18

$w h a t i s t h e r e m a i n d e r w h e n$ $767^{1009} i s d i v i d e d b y 25$
	Answered by tanmay.chaudhury50@gmail.com last updated on 01/May/18

	$s e e 25 \times 31 = 775$ ${(775 - 8)}^{1009} t h e r e i s t o t a l 1009 + 1 = 1010 t e r m s$ $t h e t e r m s c o n s i s t i n g 775 a r e d i v i s b l e b y 25$ $n o w t h e l a s t t e r m t h a t i s {(- 8)}^{1009}$ $t o b e d e v i d e d b y 25 a n d r e m a i n d e r t o f i n d$ $- (8) . 8^{1008} = - (8) . {(8^{4})}^{252}$ $= (- 8) . {(4096)}^{252}$
	Commented by tanmay.chaudhury50@gmail.com last updated on 01/May/18

	$= (- 8) {(4100 - 4)}^{252}$ $= f (25) + (- 8) {(- 4)}^{252}$ $e x c e p t l a s t t e r m (- 8) {(- 4)}^{252} r e m a i n i n g t e r m s$ $a r e d i v i s b l e b y 25$ $(- 8) {(4^{4})}^{63}$ $(- 8) {(250 + 6)}^{63}$ $t e r m s c o n s i s t i n g 250 + (- 8) {(6)}^{63}$ $e x c e p t (- 8) {(6)}^{63} r e s t d i v i s b l e b y 25$ $(- 8) {(225 - 9)}^{31}$ $= t e r m s c o n t a i n i n g 225 + (- 8) {(- 9)}^{31}$ $t e r m s c o n t a i n i n g 225 d i v i s b l e b y 25$ $(8) {(9)}^{31}$ $(8) {(10 - 1)}^{31}$ $t e r m s c o n t a i n i n g (8) (10^{31} -^{31} C 1 . 10^{30} +^{31} C_{2} . 10^{29}$ $s o r e m a i n d e r i s (8) {(1)}^{31} t h a t i s 8$
	Answered by Rasheed.Sindhi last updated on 04/May/18

	$767^{5} \equiv 7 (\mod 25)$ ${(767^{5})}^{2} \equiv 7^{2} = 49 \equiv 24 \equiv - 1 (\mod 25)$ $767^{10} \equiv - 1 (\mod 25)$ ${(767^{10})}^{2} \equiv {(- 1)}^{2} (\mod 25)$ $767^{20} \equiv 1 (\mod 25)$ ${(767^{20})}^{50} \equiv 1 (\mod 25)$ $767^{1000} \equiv 1 (\mod 25) . . . . . . . . . . . . . . . . . . . A$ $767^{3} \equiv 13 (\mod 25)$ ${(767^{3})}^{3} \equiv {(13)}^{3} = 2197 \equiv 22 (\mod 25)$ $767^{9} \equiv 22 (\mod 25) . . . . . . . . . . . . . . . . . . . . . . . . B$ $A \times B : 767^{1009} \equiv 22 (\mod 25)$
	Commented by Rasheed.Sindhi last updated on 09/May/18

	$Mr NECx pl confirm that the answer$ $is correct or not .$
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