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I think the questioner means this: given a list of integers N each of the form N=p*q with p,q prime, he/she wants to find common factors of pairs of the Ns, so to take their GCDs in pairs. This works (after assigning your list of integers to the variable Ns):
sage: gcds = [GCD(N1,N2) for N1 in Ns for N2 in Ns if N1!=N2]
sage: [g for g in gcds if g>1]
[521192137187180935658403029827,
701397335649456892851007539749,
701397335649456892851007539749,
521192137187180935658403029827]
but there are certainly more efficient ways.
