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 3: Scanning finds exactly two matches: 550 and 803

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550/11=50=5^2+5^2+0^2,\qquad 803/11=73=8^2+0^2+3^2

Detailed analysis

Running through all $81$ candidates, only $N=550$ (with $k=50=5^2+5^2+0^2$ ) and $N=803$ (with $k=73=8^2+0^2+3^2$ ) satisfy the equality; every other multiple of $11$ in the range fails, confirming the same answer obtained by the case analysis in Solution 1.