Let P(x) be polynomial in x with integral coefficients. If n is a solution of P(x)≡0(mod n) , and a≡b(mod n), prove that b is also a solution.


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Question Number 76229 by arkanmath7@gmail.com last updated on 25/Dec/19

$L e t P (x) b e p o l y n o m i a l i n x w i t h i n t e g r a l$ $c o e f f i c i e n t s . I f n i s a s o l u t i o n o f$ $P (x) \equiv 0 (m o d n), a n d a \equiv b (m o d n),$ $p r o v e t h a t b i s a l s o a s o l u t i o n .$
	Answered by Rio Michael last updated on 25/Dec/19

	$g e n e r a l l y i f a \equiv b (m o d n) t h e n n ∣ (a - b)$ $i f n ∣ (a - b) a n d n ∣ p (x) f r o m p (x) \equiv 0 (m o d n)$ $t h e n (a - b) i s a l s o a s o l u t i o n$ $n o w i f a \equiv b (m o d n) t h e n b \equiv a (m o d n)$ $w h i c h m e a n s a \equiv b (m o d n) h e n c e a a n d b$ $a r e s o l u t i o n s t o p (x) \equiv 0 (m o d n)$ $h e n c e b i s a s o l u t i o n .$
	Commented by arkanmath7@gmail.com last updated on 25/Dec/19

	$t h n x s i r$
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