I have a late 2013 iMac and have purchased an M1 Mac mini. Can the iMac be used as the monitor ?

Sorry, none of the new Mac's including the Mac mini support Target Display Mode.

see > Use your iMac as a display with target display mode - Apple Support

You will need an external display to setup and use a Mac mini.

see > Connect a display to Mac mini - Apple Support

I have a 2013 iMac and an M1 Mac and and verify that there is no adapter, gadget, software or firmware hack, or magic powders that can make it work as a display with an M1 Mac.

The first article I posted, is saying that Target Display Mode is no longer supported with any cable.

please carefully read > Use your iMac as a display with target display mode - Apple Support