MathLabs

Problem 1

Find the smallest natural number whose decimal representation ends in 66 and for which moving this final digit to the front produces four times the original number.
Step 4 of 5: Find the least possible digit length
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10,9,12,3,4(mod13)for m=1,2,3,4,510,9,12,3,4\pmod{13}\quad\text{for }m=1,2,3,4,5
Detailed analysis

The successive residues of 10m10^m modulo 1313 for m=1,2,3,4,5m=1,2,3,4,5 are 10,9,12,3,410,9,12,3,4. Thus m=5m=5 is the first possible value (and m=0m=0 cannot describe the required rotated multi-digit number).