MathLabs

Problem 2

Find an integer nn in the range 100≤n≤1997100\le n\le1997 such that nn divides 2n+22^n+2.
Step 1 of 4: Find useful prime congruences
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23≡−2(mod5),26≡−2(mod11)2^3\equiv-2\pmod5,\qquad2^6\equiv-2\pmod{11}
Detailed analysis

For primes, Fermat's theorem makes powers periodic. The official search records 23≡−2(mod5)2^3\equiv-2\pmod5 and 26≡−2(mod11)2^6\equiv-2\pmod{11}, then searches compatible products.