How many of 1,2,3,4,5...11000 are invertible modulo 880? My work: I write 880 as $2^4$ x $5$ x $11$. I know that for a number to be invertible modulo 880, it must be coprime to 880. I have to count all such numbers from 1,2,3.....879 which I can then extrapolate to 11000. How do I proceed from here? I'm supposed to do this without a calculator

You are looking for $\phi(880)$ where $\phi$ is Euler's totient function. Half the numbers are odd, $\frac 45$ are not divisible by $5$, and $\frac {10}{11}$ are not divisible by $11$, so the fraction coprime to $880$ is $\frac 12 \cdot \frac 45 \cdot \frac {10}{11}$