A Pythagorean triplet is a set of three natural
numbers, a < b < c, for which,
For example, 32 + 42 = 9 + 16 = 25 = 52.
There exists exactly one Pythagorean triplet for
which a + b + c = 1000.
Find the product abc.
There are three variables and two equations. We can reduce this to two variables and 1 equation. After that we will have to switch to a numerical approach with the added constraint that the results be integers greater than 0.
Start with
a2+b2=c2 - (1) and
a+b+c=1000 (2)
(2) gives c=1000-(a+b)
Substituting this in (1) gives
a2+b2=[1000-(a+b)]2 -- (3)
Simplifying (3) yields
2000*(a+b) - 2*a*b = 106
Designate two cells in some worksheet as representing a and b. I chose G2 and G3. Then, in some other cell enter the formula =2000*(G2+G3)-2*G2*G3-10^6. Create a Solver model that specifies the cell with the formula should have a value of zero and that the variables be integer values greater than zero. The Solver solution will be the answer.
