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It is well-known that given two primes $p$ and $q$, $pZ + qZ = Z$ where $Z$ stands for all integers. It seems to me that the set of natural number multiples, i.e. $pN + qN$ also span all natural numbers that are large enough. That is, there exists some $K>0$, such that $$pN + qN = [K,K+1,...).$$

My question is, given $p$ and $q$, can we get a upper bound on $K$?

  • 0
    I assume you mean pN + qN contains [K, K+1, ...). It's not hard to see that below K there are gaps.2010-10-28
  • 5
    This works the same whether p and q are prime or not, as long as their greatest common divisor is 1.2010-10-28

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