In a puzzle game, three friends - Alice, Bob, and Carol - each make a statement about a mysterious number, \( N \), which is an integer between 1 and 100. 
- Alice says: "The number is a power of 2."
- Bob says: "The number is odd."
- Carol says: "The number is a multiple of 5."

Only one of these statements is true. What is \( N \)? Let's think step by step and output the final answer within \boxed{}.
user