FragmentedG 30

As the achievement states, you must fully complete one fragment section. The Quiz Mastery section may be an easy way out but, I think doing the gravitational cores is another route to take. If some of them don't appear, progress to the end of chapter 4 where they will end up giving you a mission to find all 7 which will unlock all 7 cores. Also, if you try to get some of them before the mission pops up, you need Mog's ability to make things invisable, visable. Plus you must collect 5(not 7) cores to progress with the game anyways. With having to find 5 regardless, you might as well get the other 2 to finish the section completely.

Edit: Actually you only need 5 to progress the story and not 7. Thanks Torash for pointing that out.

Here are them 7 cores and their locations

Core name and Location:
#1 - Gravitation Core Alpha - New Bodhum (003AF)
#2 - Gravitation Core Beta - Bresha Ruins (005AF)
#3 - Gravitation Core Gamma - Oerba (200AF)
#4 - Gravitation Core Delta - Academia (400AF)
#5 - Gravitation Core Epsilion - Yaschas Massif (100AF)
#6 - Gravitation Core Zeta - Oerba (400AF)
#7 - Gravitation Core Eta - Sunleth Waterscape (400AF)
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HOW TO GET TO EACH AREA:
#1 New Bodhum 003AF - It's the area where the game starts. Return later to collect the core (it won't be there in the beginning)
#2 Bresha Ruins 005AF - Second area you enter during the story. No way to miss it.
#3 Oerba 200AF - Oerba 200AF - from Bresha Ruins go through orange gate to Yaschas Massif 10AF and from there through orange gate to Oerba 200AF
#4 Academia 400AF - You will get here as you progress in the story
#5 Yaschas Massif 100AF - through blue gate in Academia 400AF (requires wild artefact)
#6 Oerba 400AF - from blue portal in augusta tower 49th floor (requires wild artefact) travel to Oerba 300AF. Solve all paradoxes in Oerba 300AF and go throug the blue gate (requires wild artefact) to reach Oerba 400AF
#7 Sunleth Waterscape 400AF - through blue gate in Yaschas Massif 100AF (requires wild artefact). Once in waterscape you will have to search for the missing pudding monsters to reach the core.