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August 11th, 2016, 07:09 PM   #1
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Question All primes are in form of 10nk+c

Prove:

For positive integer n, and any positive integer c that 0 < c < 10n and c is coprime with 10n, all primes larger than 10n can be expressed in form of 10nk+c where k is a positive integer.

Remark: c may have various values. For example, when n=2, possible values of c are: 1,3,7,9,11,13,17,19
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August 11th, 2016, 11:53 PM   #2
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Let $m = nk$. Then setting $n = 1$, it is clear that $m$ can be any natural number.

It is a consequence of the division algorithm that any positive number (including primes) can be represented by $10m + c$, where $0 < c < 10$. This proves the result.
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August 12th, 2016, 08:31 AM   #3
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But if $c$ is coprime to $10n$ how would you represent, say, 12?
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August 12th, 2016, 08:59 AM   #4
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If $c$ isn't coprime to $10n$, $10n + c$ isn't a prime.
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August 12th, 2016, 03:26 PM   #5
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Ah. I didn't read the question very well it seems.

It is still trivial however, since 1, 3, 7 and 9 are all coprime to $10n$.
This means that all odd numbers not ending in 5 (and thus not divisible by 5) can be represented in this way.
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