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Say I have some integer n in some interval such that,

gcd(n, k) = gcd(n + 1, k) = 1, for some composite odd integer k >= 9

I want to know if such n exists in that interval. To know that one exists suffices.

I have tried to think in terms of modular arithmetic where for all primes in k, the smaller of which is "embedded" inside the larger one: e.g. given two primes 5 and 13, the latter determines the outer "clock" while, begining from zero, number 5 puts marks on that clock (which uses arithmetic modulo 13). For some obvious reasons, this got really complicated...

I'm not sure either whether writing out a linear combination for each gcd will lead somewhere.

I'm not expecting a full solution. I just need some guidance (or to know whether this is solvable at all).

Thanks.

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# B Consecutive integers, each relatively prime to some k

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