The RSA algorithm for public-key cryptography is based on the presumed difficulty of factoring large bi-prime integers, the factoring problem. However, as pointed out in Lenstra et al [1] it is possible, given a set of moduli, to factor some of them by finding shared primes. This way the factoring problem is bypassed using a simple greatest common divisor (gcd) operation.
An example implementation can be found here.
An example implementation can be found here.
Read more: RSA sanity check: Testing RSA moduli for shared prime factors