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 3 of 4: Exclude part (b)
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2n≢6(mod7)2^n\not\equiv6\pmod7
Detailed analysis

The cycle contains only 2,4,12,4,1, never 6≡−1(mod7)6\equiv-1\pmod7. Thus 2n+1≢0(mod7)2^n+1\not\equiv0\pmod7 for every positive integer nn.