Mark the Correct alternative in the following:

For all n ∈ N, 3 × 5^{2n + 1} + 2^{3n + 1} is divisible by

If P(n): 2 × 4^{2n + 1} + 3^{3n + 1} is divisible by λ for all n ∈ N is true, then find the value of λ.

Mark the Correct alternative in the following:

If xⁿ – 1 is divisible by x - λ, then the least positive integral value of λ is

Mark the Correct alternative in the following:

If 10ⁿ + 3 × 4^{n + 2} + λ is divisible by 9 for all n ∈ N, then the least positive integral value of λ is

Mark the Correct alternative in the following:

Let P (n): 2n < (1 × 2 × 3 × … × n). Then the smallest positive integer for which P(n) is true is