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 3 of 5: Extract the divisibility condition
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n=2n′⟹13n′=10m−4n=2n'\qquad\Longrightarrow\qquad 13n'=10^m-4
Detailed analysis

The equation shows that nn is even, so write n=2n′n=2n'. It follows that 13n′=10m−413n'=10^m-4, or equivalently 10m≡4(mod13)10^m\equiv4\pmod{13}.