MathLabs

Problem 1

(a) Find all positive integers nn for which 2n−12^n-1 is divisible by 77. (b) Prove that no positive integer nn makes 2n+12^n+1 divisible by 77.
Step 1 of 4: Find the residue cycle
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21≡2,22≡4,23≡1(mod7)2^1\equiv2,\quad2^2\equiv4,\quad2^3\equiv1\pmod7
Detailed analysis

Since 23=8≡1(mod7)2^3=8\equiv1\pmod7, multiplying by 22 repeats the cycle 2,4,12,4,1 for all positive exponents.