If p is any Prime number, then p12 would meet the requirements. The seven factor pairs would be:
(p0, p12)
(p1, p11)
(p2, p10)
(p3, p9)
(p4, p8)
(p5, p7)
(p6, p6)
It can be shown that, because 7 is a prime, a composite number cannot generate such a solution.
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(1, 21) (3, 7)
There are 3 factor pairs of 245:245 = 1 x 245245 = 5 x 49245 = 7 x 35
The factor pairs for 77 are (1,77) and (7,11).
They are: 1, 2, 7 and 14
14 and 21, and 21 and 28 are the number pairs between 15 and 30 that have 7 as their greatest common factor.