Discussion
Which of the following natural numbers cannot be written as a sum of 2 squares ?
Answer: B
The number that is not a sum of two squares is 1111.
A number is a sum of two squares exactly when every prime \(\equiv 3 \pmod 4\) in its factorisation appears to an even power. \(1111 = 11 \times 101\), and 11 \(\equiv 3 \pmod 4\) appears once. A quicker check: \(1111 \equiv 3 \pmod 4\), and a sum of two squares is 0, 1 or 2 mod 4.
The others are sums of two squares: \(405 = 18^{2} + 9^{2}\), \(117 = 9^{2} + 6^{2}\), and \(164 = 10^{2} + 8^{2}\).
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