2^ 3 n -7 n-1 is divisible by

Step-by-step solution

Let f(n) = 2^ 3n - 7n - 1 = 8^n - 7n - 1 . Using the binomial expansion, 8^n = (1 + 7)^n = 1 + ^ n C₁(7) + ^ n C₂(7^2) + ^ n C₃(7^3) + + ^ n C_ n (7^n) . Substituting this into the expression for f(n) : f(n) = (1 + 7n + ^ n C₂(49) + ^ n C₃(7^3) + + 7^n) - 7n - 1 . f(n) = ^ n C₂(49) + ^ n C₃(7^3) + + 7^n . Since every term in the expansion contains at least 49 as a factor, f(n) is divisible by 49 . Answer: 49