2011 UNCO Math Contest II Problems/Problem 8

Problem

The integer $45$ can be expressed as a sum of two squares as $45 = 3^2 + 6^2$ .

(a) Express $74$ as the sum of two squares.

(b) Express the product $45\cdot 74$ as the sum of two squares.

(c) Prove that the product of two sums of two squares, $(a^2+b^2)(c^2+d^2)$ , can be represented as the sum of two squares.

(a) By testing the perfect squares up to $74$ , you can see that $\boxed{74=25+49=5^2+7^2}$ . (b) $45 \cdot 74 = 9^2 + 57^2 = 27^2 + 51^2$ .

2011 UNCO Math Contest II (Problems • Answer Key • Resources)
Preceded by Problem 7	Followed by Problem 9
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