how can i install purchased Photoshop cs5 on macbook pro M1?

Ever get it installed?

No way possible unless you purchase Parallels for Mac (80 Euros/year) and install Mojave on Parallels virtual machine.  That's assuming you have a valid CS5 installation disk and serial number for Mac.  If you have Windows software, you'll need to install Windows 10 on Parallels.

The other option is to buy a Creative Cloud plan for your new Monterey machine.

Creative Cloud Photography Plan (approx $10/month for 12 months). Includes the following:
- Photoshop CC and Photoshop on iPad
- Lightroom on desktop, mobile and the web
- Lightroom Classic
- Portfolio website + hosting
- Spark with premium features
- Bridge
- 20 GB cloud storage (upgradable to 1 TB).
https://www.adobe.com/creativecloud/photography.html

Creative Cloud All Apps (51 + apps & services) and Single Apps Plans
https://www.adobe.com/creativecloud/plans.html

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There is no way to install CS5 on an M1 running Monterey.

You'll have to dual boot with CS5 on an external hard drive running Mojave or earlier. [Update: Not an option for an M1 Mac]

Or sign up to a Cloud plan and run the current version of Photoshop natively on your M1.

You could try a Cloud plan for free for 7 days. If you choose not to purchase you MUST cancel the trial before the 7 days expires or it will automatically sign you up to a paid plan.