the same serial number will work.
you need, however, to use the same suite used to obtain that serial number. ie, it's valid for that suite not the stand-alone indesign program.
start the suite setup and you will be given the option to install id only. you do not need to reinstall the entire suite.
if you need the installation files:
Downloads available:
- Suites and Programs: CC 2014 | CC | CS6 | CS5.5 | CS5 | CS4 | CS3
- Acrobat: XI, X | 9,8 | 9 standard
- Premiere Elements: 13 | 12 | 11, 10 | 9, 8, 7
- Photoshop Elements: 13 |12 | 11, 10 | 9,8,7
- Lightroom: 5.6| 5 | 4 | 3
- Captivate: 8 | 7 | 6 | 5
- Contribute: CS5 | CS4, CS3
Download and installation help for Adobe links
Download and installation help for Prodesigntools links are listed on most linked pages. They are critical; especially steps 1, 2 and 3. If you click a link that does not have those steps listed, open a second window using the Lightroom 3 link to see those 'Important Instructions'.