Question: "Prove 10P(n)2+1 is divisible by n"

Oxinailelpels3t14

Answered question

2022-04-01

Prove

10^{\frac{P (n)}{2}} + 1

is divisible by n

Answer & Explanation

Wilson Rivas

Beginner2022-04-02Added 12 answers

Step 1
Consider $n = 21$ then
$\frac{1}{21} = 0 . \overset{―}{047619}$
thus $P (21) = 6$ . However
$10^{3} + 1 \equiv 14 (\mod 21)$
Step 2
Another example is $n = 13 \cdot 17 = 221$ , then $P (221) = 48$ and
$10^{24} + 1 \equiv 119 (\mod 221)$
Step 3
However, the statement is true if n is ', different from 2 or 5. This is because $P (n) = or d_{n} (10)$ (from the definition of multiplicative order, also see this post and this). From (1) in your post
$10^{P (n)} \equiv 1 (\mod n) \Rightarrow n | (10^{\frac{P (n)}{2}} + 1) \cdot (10^{\frac{P (n)}{2}} - 1)$
Using Euclid's lemma, if we assume $n ∣ 10^{\frac{P (n)}{2}} - 1$ , then $\frac{P (n)}{2} \geq or d_{n} (10) = P (n)$ which is a contradiction. As a result
$n ∣ 10^{\frac{P (n)}{2}} + 1$

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