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A Pythagorean triplet is a set of three natural
numbers, `a` < `b` < `c`, for which,

For example, 3^{2}
+ 4^{2}
= 9 + 16 = 25 = 5^{2}.

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

a^{2}+b^{2}=c^{2} - (1) and

a+b+c=1000 (2)

(2) gives c=1000-(a+b)

Substituting this in (1) gives

a^{2}+b^{2}=[1000-(a+b)]^{2} -- (3)

Simplifying (3) yields

2000*(a+b) - 2*a*b = 10^{6}

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.