69x ≡ 1 (mod 31) solve for x


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Question Number 96928 by bobhans last updated on 05/Jun/20

$69 x \equiv 1 (\mod 31)$ $solve for x$
Commented by RAMANA last updated on 05/Jun/20

$s e n d t h e a n s w e r p l e a s e$
	Answered by MAB last updated on 05/Jun/20

	$69 x \equiv 1 [31]$ $69 x - 31 \times 2 \times x \equiv 1 [31]$ $7 x \equiv 1 [31]$ $w e h a v e 31 i s p r i m e a n d 7 x \equiv 1 [31]$ $a n d t h e o n l y i n v e r s e o f 7 i n t h e c l a s s 31$ $i s 9 (7 \times 9 = 63 = 2 \times 31 + 1)$ $t h e r f o r e x \equiv [31]$
	Answered by 1549442205 last updated on 06/Jun/20

	$we have 69 x - 1 = 31 a (a \in Z) \Leftrightarrow a = \frac{69 x - 1}{31} = 2 x + \frac{7 x - 1}{31}$ $\Rightarrow 7 x - 1 = 31 b (b \in Z) \Leftrightarrow x = \frac{31 b + 1}{7} = 4 b + \frac{3 b + 1}{7}$ $\Rightarrow 3 b + 1 = 7 c (c \in Z) \Leftrightarrow b = \frac{7 c - 1}{3} = 2 c + \frac{c - 1}{3}$ $\Rightarrow c - 1 = 3 d (d \in Z) \Rightarrow c = 3 d + 1 . From this we get$ $b = 2 c + d = 7 d + 2 \Rightarrow x = 4 b + c = 31 d + 9$ $Thus, x = 31 d + 9 or x = 9 (\mod 31)$
	Answered by mr W last updated on 06/Jun/20

	$69 x = 31 n + 1$ $69 x - 31 n = 1$ $\Rightarrow x = 31 k + 9 \Rightarrow x \equiv 9 m o d 31$ $s e e a l s o Q 44704, Q 19198$
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