Cutting random pieces of string

Probability Level 3

From a string of length $1 \text{ m},$ we choose a length uniformly at random between $0 \text{ m}$ and $1 \text{ m},$ and cut as many of these lengths as possible from the string.

What is the expected length of the remaining string (in meters)?

Bonus: If we have a random number $a$ chosen with uniform distribution over the interval $\big(0,\frac{1}{n}\big],$ where $n\in\mathbb{N}$ , what's the expected value of the remainder $1\bmod a$ for a particular $n?$