The key on a piano corresponding to the note of freqency 880

The key on a piano corresponding to the note of freqency 880Hz is depressed very gently,so that the hammer does not strike the string,but the damper is lifted and the string is free to vibrate.When the key corresponding to the note of frequency 440Hz is struck firmly,it is found that the 880Hz string also vibrates.Give a brief explanation of this phenomenon.Suppose the 220Hz string had also been free to vibrate.What frequencies,if any,would it have emitted when the 440 Hz key was struck?