Problem 4

The positive integers

a

and

b

are such that

15a+16b

and

16a-15b

are both squares of positive integers. What is the least possible value of the smaller of these two squares?

Step 3 of 4: Use residues modulo

37

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In plain words

Use residues modulo $37$

m\equiv n\equiv0\pmod{37}

Detailed analysis

Modulo $37$ , the same identity gives $m^2/n^2\equiv31$ when $n\not\equiv0$ . The residue $31$ is a quadratic nonresidue modulo $37$ , so again $n\equiv m\equiv0\pmod{37}$ . Thus $481\mid m,n$ .