Find all positive integers n

Number Theory Level 3

For which positive integers $n$ is the number $3^n + 1$ a multiple of $10$ ?

Note: $k$ in the answer options denotes a non-negative integer.

n = 2k

n = 4k + 2

n = 6k - 4

n = 3k + 2

1 solution

Chew-Seong Cheong
Feb 16, 2019

When $3^n+1$ is a multiple of $10$ , we have:

$\begin{aligned} 3^n + 1 & \equiv 0 \text{ (mod 10)} \\ \implies 3^n & \equiv 9 \text{ (mod 10)} \\ (10-1)^{2k}\cdot 9 & \equiv 9 \text{ (mod 10)} \\ 3^{4k} \cdot 3^3 & \equiv 9 \text{ (mod 10)} \\ 3^{4k+2} & \equiv 9 \text{ (mod 10)} \end{aligned}$

Therefore, $\boxed {n=4k+2}$ .

Find all positive integers n

1 solution

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