Photoshop. Open the design in Photoshop. Once open, there will be a tab for the image at the top of the page. The colour mode will be displayed at the end of the tab. In this example, we can see that this 25mm badge design is set up in CMYK mode.
Illustrator. Illustrator displays the colour mode in the exact same way. Open the design in Illustrator and then check the tab at the top. In this example we can see that the image is set up in RGB colours.