From 1 to 100!

Let $n$ be the unknown number.

Since $n$ has 7 factors $\Rightarrow$ $n = k^{7-1} = k^6$ for some integer $k \geq 1$ .

Since $1 \leq n \leq 100 \Rightarrow 1 \leq k \leq 2$ because $3^6 = 729 >100$ .

Case 1: If $k = 1 \Rightarrow n = 1^6 = 1$ , but 1 has only 1 factor, just 1.

Case 2: If $k = 2 \Rightarrow n = 2^6 = 64$ .

Since 64 has factors $2^0, 2^1, 2^2, \ldots, 2^6 \Rightarrow n = \boxed{64}$ .

Click here to see my code based solution to this question. On the bottom, you can see that the program searched and found the number 64 with 7 factors and printed it out with its factors.

$64=2^6$

Therefore, the factors are $2^0 (1) , 2^1 (2) , 2^2 (4) ... 2^6 (64)$