Problem 1

Determine all three-digit numbers

N

that are divisible by

11

, such that

N/11

equals the sum of the squares of the digits of

N

Step 3 of 6: Case A: parity forces b to be a multiple of 4

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b=2B \implies B=a(a-5)+2aB+4B^2 \text{ is even} \implies b\in\{0,4,8\}

Detailed analysis

From $2a^2+2ab+2b^2=10a+b$ , the left side is even, so $b$ is even; write $b=2B$ . Substituting and simplifying shows $B=a(a-5)+2aB+4B^2$ , whose right-hand side is even, so $B$ itself is even. Hence $b=2B$ is a multiple of $4$ , leaving only $b=0,4,8$ to check.