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An example of a problem which is known to be in NP and in co-NP is integer factorization: given positive integers m and n determine if m has a factor less than n and greater than one.
Membership in NP is clear ; if m does have such a factor then the factor itself is a certificate.
Membership in co-NP is also straightforward: one can just list the prime factors of m, which the verifier can confirm to be valid by multiplication and the AKS primality test.

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