# 1, 5, 2, and 3

$$(1,5,2,3)$$ are 4 distinct positive integers such that the sum of 2 of them equals the product of the other 2, and vice versa, i.e. the sum of these 2 other numbers equals the product of the 2 numbers initially mentioned: $1+5=2\times 3, \quad 2+3=1\times 5.$ Generally spoken, $a+b=c\times d, \quad c+d=a\times b.$ Determine the number of un-ordered quadruplets $$(a,b,c,d)$$ other than $$(1,5,2,3)$$ that satisfy this condition.

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