MathLabs

Problem 1

Determine all three-digit numbers NN that are divisible by 1111, such that N/11N/11 equals the sum of the squares of the digits of NN.
Step 1 of 3: Bounding the candidates: 3-digit multiples of 11
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N∈{110,121,…,990}, 81 candidatesN \in \{110,121,\dots,990\},\ 81 \text{ candidates}
Detailed analysis

The three-digit multiples of 1111 are exactly 11×10,11×11,…,11×9011\times10, 11\times11,\dots,11\times90, i.e. 8181 numbers from 110110 to 990990. It suffices to test the digit-square condition on each.