Competition problem — Number theory

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Number theory Difficulty 5.5 AIME, harder Find the answer

3. Find all possible representations as a sum of two squares for (i) $5 \cdot 13$ , (ii) $17 \cdot 29$ , (iii) $37 \cdot 41$ , (iv) $5 \cdot 13 \cdot 17 \cdot 29$ , (v) $7^{2} \cdot 13 \cdot 17$ .

Your answer

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Solution

$\begin{array}{l}\text { 3. (i) } 5 \cdot 13=8^{2}+1^{2}=7^{2}+4^{2} \text {; (ii) } 17 \cdot 29=22^{2}+3^{2}=18^{2}+13^{2} \text {; (iii) } 37 \\ 41=34^{2}+19^{2}=29^{2}+26^{2} \text {; (iv) } 5 \cdot 13 \cdot 17 \cdot 29=179^{2}+2^{2}=173^{2}+46^{2}=178^{2}+ \\ 19^{2}=163^{2}+74^{2}=157^{2}+86^{2}=131^{2}+122^{2}=166^{2}+67^{2}=142^{2}+109^{2} \text {; (v) } 7^{2} \\ 13 \cdot 17=(7 \cdot 14)^{2}+(7 \cdot 5)^{2}=(7 \cdot 11)^{2}+(7 \cdot 10)^{2}\end{array}$

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.