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Question Number 57075 by Tawa1 last updated on 30/Mar/19

Commented by Tawa1 last updated on 30/Mar/19

$Please i want to understand how the - 1 in the expansion becomes - 13$ $and the + 60 outside . and how the remainder is 8 . Using the same method .$
Commented by 121194 last updated on 30/Mar/19

$- 1 = 12 - 13$ $5 \times 12 = 60$ $since$ $13 ∣ 26$ $then$ $5 \times (- 13) + 52 + 8$ $13 k + 52 + 8$ $and$ $13 ∣ 52$ $so you can easy deduct that 5^{99} % 13 = 8$
Commented by Tawa1 last updated on 30/Mar/19

$God bless you sir . I appreciate$
Commented by JDamian last updated on 30/Mar/19

$5^{99} = 5 {(26 - 1)}^{49}$ $(5^{99}) m o d 13 = [5 {(26 - 1)}^{49}] m o d 13 =$ $= (5 m o d 13) \times {(26 - 1)}^{49} m o d 13 =$ $= 5 \times {(26 m o d 13 - 1 m o d 13)}^{49} =$ $= 5 \times {(0 - 1)}^{49} = - 5$ $- 5 m o d 13 = (- 5 + 13) m o d 13 = 8 m o d 13 =$ $= 8$
Commented by Tawa1 last updated on 30/Mar/19

$God bless you sir . I appreciate .$
	Answered by tanmay.chaudhury50@gmail.com last updated on 30/Mar/19
	$5^(99) (5^3 )^(33) (125)^(33) (117+8)^(33) (9×13+8)^(33) (9^ ×13)^(33) +33c_1 (9×13)^(32) (8)+...+8^(33) f(13)+(8^3 )^(11) f(13)+(507+5)^(11) f(13)+(39×13+5)^(11) f(13)+{(39×13)^(11) +11c_1 (39×13)^(10) ×5+...+5^(11) } f(13)+g(13)+5^(11) f(13)+g(13)+(5^3 )^3 (5^2 ) f(13)+g(13)+(117+8)^3 (25) f(13)+g(13)+25[(13×9)^3 +3(13×9)^2 ×8+3(13×9)×8^2 +8^3 ] f(13)+g(13)+25h(13)+25×8^3 now calculating 25×8^3 (13+12)(507+5) (13+12)(39×13+5) 13×39×13+13×5+12×39×13+5×12 ϕ(13)+(4×13+8) so remainder is 8$
	$5^{99}$ ${(5^{3})}^{33}$ ${(125)}^{33}$ ${(117 + 8)}^{33}$ ${(9 \times 13 + 8)}^{33}$ ${(9^{} \times 13)}^{33} + 33 c_{1} {(9 \times 13)}^{32} (8) + . . . + 8^{33}$ $f (13) + {(8^{3})}^{11}$ $f (13) + {(507 + 5)}^{11}$ $f (13) + {(39 \times 13 + 5)}^{11}$ $f (13) + {{(39 \times 13)}^{11} + 11 c_{1} {(39 \times 13)}^{10} \times 5 + . . . + 5^{11}}$ $f (13) + g (13) + 5^{11}$ $f (13) + g (13) + {(5^{3})}^{3} (5^{2})$ $f (13) + g (13) + {(117 + 8)}^{3} (25)$ $f (13) + g (13) + 25 [{(13 \times 9)}^{3} + 3 {(13 \times 9)}^{2} \times 8 + 3 (13 \times 9) \times 8^{2} + 8^{3}]$ $f (13) + g (13) + 25 h (13) + 25 \times 8^{3}$ $n o w c a l c u l a t i n g 25 \times 8^{3}$ $(13 + 12) (507 + 5)$ $(13 + 12) (39 \times 13 + 5)$ $13 \times 39 \times 13 + 13 \times 5 + 12 \times 39 \times 13 + 5 \times 12$ $φ (13) + (4 \times 13 + 8)$ $s o r e m a i n d e r i s 8$
	Commented by peter frank last updated on 30/Mar/19

	$n i c e w o r k$
	Commented by Tawa1 last updated on 30/Mar/19

	$God bless you sir . I appreciate$
	Commented by tanmay.chaudhury50@gmail.com last updated on 30/Mar/19

	$t h a n k y o u . . . i t r i e d t o m a k e y o u u n d e r s t a n d . . .$
	Commented by peter frank last updated on 30/Mar/19

	$t h a n k s a g a i n$
	Commented by malwaan last updated on 31/Mar/19

	$thank you$
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