If a, b,c and d are natural numbers such that a^2+b^2=41 and c^2+d^2=25 , then the polynomial whose squares are (a+b) and (c+d) can be
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Suppose that a is less than equal to b and c is less than equal to d. Since swapping a and b or c and doesn't change any expression a^2 + b^2, c^2 +d^2, (a+b), (c+d)
Hence, a^2 is less than 41/2 means a = 0, 1, 2, 3, or 4 and a^2 = 0, 1, 4, 9, 16
b^2 = 41 - a^2 = 25
c^2 is less than 25/2 means c=0, 1, 2, 3 and c^2 = 0, 1, 4, 9
d^2=25-c^2=25 or 16
then {a,b}={4,5} and {c,d}={0,5} or {3,4}
so (a+b) = 9 and c+d =5 or 7
Hence, a^2 is less than 41/2 means a = 0, 1, 2, 3, or 4 and a^2 = 0, 1, 4, 9, 16
b^2 = 41 - a^2 = 25
c^2 is less than 25/2 means c=0, 1, 2, 3 and c^2 = 0, 1, 4, 9
d^2=25-c^2=25 or 16
then {a,b}={4,5} and {c,d}={0,5} or {3,4}
so (a+b) = 9 and c+d =5 or 7
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