Product makes the sums

[BMAD]

Does their exist four distinct natural numbers (a,b,c,d) where a*b, c*d is equivalent to the sum of all four numbers?

 

What if i add the condition that a*d also equals the sum of all four numbers?

[DejMar]

There are two sets of four distinct natural numbers {a,b,c,d} where a*b = c*d = a+b+c+d: {2,15,3,10} and {2,12,4,6}. This is, of course, in addition to the sets of the same numbers ordered differently, i.e., swapping values between the mutliplicands of one expression.or the swapping of the pair with that of the other pair, such as (15, 2, 10, 3).

There is no set of four distinct natural numbers where a*b = c*d = a*d = a+b+c+d, as a*d = c*d would mean that a = c, and a*b = a*d would mean b = d, both of which contradict the given that the natural numbers are distinct. 

Edited January 20, 2015 by DejMar