The GCD is
, so
divides both
and
. For some
and
we can write
Expanding the right sides, we get
also divides the LCM, so
and expanding gives
Matching up all the coefficients, we have
All the unknowns are integers, and we have the prime factorization 6 = 2×3 that also satisfies 5 = 2 + 3.
• If
and
, then
,
, and
.
• If
and
, then there is no solution for
since
but 3 ≠ 2 - 1 = 1.
It follows that
.