Question

If 3³ⁿ - αn - β is divisible by 676 for all n ∈ N, then (α, β) is, α, β ∈ Z

Answer 1

3³ⁿ - αn - ß is divisible by 676 for all n \(\in\) N

(a) if α  = -26, ß = -1

then 3³ⁿ -αn - ß = 3³ⁿ + 26n + 1 = f₁(n) (Let)

for n = 1, f₁(1) = 3³ + 26 + 1 = 54 not divisible by 676

(b) if α = 26, ß = 1

then 3³ⁿ - αn - ß = 3³ⁿ - 26n - 1 = f₂(n) (Let)

f₂(1) = 3³ - 26 - 1 = 27 - 27 = 0 divisible by 676.

f₂(2) = 3⁶ - 52 - 1 = 729 - 53 = 676 divisible by 676

Hence, (α, ß) = (26, 1)

(c) if α = 1, ß = 26

then 3³ⁿ  - αn - ß = 3³ⁿ - n - 26 = f₃(n) (Let)

f₃(2) = 729 - 27 = 702 not divisible by 676

(d) if α = -1, ß = -26

then 3³ⁿ - αn - ß = 3³ⁿ + n + 26 = f₄(n) (Let)

f₄(1) = 3³ + 1 + 26 = 27 + 27 = 54 not divisible by 676.