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Determine all possible quadruplet(s) (A, B, C, D) of positive integers, with A >= C, that satisfy this system of equations:

A*B = C + 2*D, and:

C*D = A + 2*B

Prove that no other quadruplet(s) satisfy the above system of equations.

Edited by K Sengupta
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Let a = nc

Let b = md

Then;

mncd = c + 2d

cd = nc + 2md

Combining both equaitons,

mn²c + 2m²nd = c + 2d

This means that both m and n must be 1

so, a = c and b = d

This gives the equation as

cd = c + 2d

let c = xd

Then, xd² = d(x+2)

x(d-1) = 2

This is possible when x = 1 or x = 2

Then, d = 3 and d = 2 respectively

Accordingly, the quadruplets (a,b,c,d) are,

(4,2,4,2) and (3,3,3,3)

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