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 2 of 6: Case A (no carry): write the digit-square condition
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a2+(a+b)2+b2=10a+b(no carry: a+b≤9)a^2+(a+b)^2+b^2=10a+b \quad(\text{no carry: } a+b\le 9)
Detailed analysis

When a+b≤9a+b\le 9, the digits of NN are exactly a, a+b, ba,\ a+b,\ b, so the required equality N/11=N/11= sum of squares of the digits reads 10a+b=a2+(a+b)2+b210a+b=a^2+(a+b)^2+b^2. Expanding the right side gives 2a2+2ab+2b2=10a+b2a^2+2ab+2b^2=10a+b.