For how many integers n is 1^n = n^1?

A. none
B. 1
C. 2
D. 3
E. More than 3

The OA is the option B.

How can I show that there is no more than 1 integer? Experts, can you show me this? Thanks.[/spoiler]

Hi M7MBA,
Let's take a look at your question.

We are asked to find, for how many integers,
$$1^n=n^1$$
Let's try out some random values.
For n = -1
$$1^{(-1)}=(-1)^1$$
$$\frac{1}{1}=-1$$
$$1\ne-1$$

For n = 0
$$1^0=(0)^1$$
$$1\ne0$$

For n = 1
$$1^1=(1)^1$$
$$1=1$$
Therefore, 1^n=n^1 is true for n = 1

Let's check for n = 2,
$$1^2=(2)^1$$
$$1\ne2$$

The given equation will only be true for n = 1
Hence, Option B is correct.

Hope it helps.
I am available if you'd like any follow up.