We need to ask ourselves what is being modified by the phrase. Is the phrase describing a noun or an action?
1) According to its designers, a recently improved type of technological device could, by shutting down potentially dangerous reactions within lithium-ion batteries, prevent laptops and mobile phones from catching fire - here 'by..batteries' is an adverbial modifier
The phrase by...batteries is describing how the improved type of device could prevent laptops from catching fire. Thus, the phrase is functioning as an adverb because it is telling us more about an action (the verb could prevent).
2) By shutting down potentially dangerous reactions within lithium-ion batteries, the designers of a recently improved type of technological device say it could prevent laptops and mobile phones from catching fire - here 'by..batteries' is an adjective modifier
The phrase by...batteries cannot be functioning as an adjective because it is not describing a noun.
What kind of designers? The designers by shutting down potentially dangerous reactions. This sentence doesn't make sense.
What kind of device? A device by shutting down potentially dangerous reactions. This sentence doesn't make sense.
The phrase by...batteries must still be functioning as an adverb. How does the type of device prevent laptops from catching fire? By shutting down potentially dangerous reactions. This sentence makes sense. So the phrase by...batteries is an adverb modifying could prevent.
Please note that the sentence above is unlikely to be the correct answer on the GMAT because the adverb by batteries is placed quite far from the verb that it's modifying.
3) A recently improved type of technological device could prevent laptops and mobile phones from catching fire, according to its designers, with it shutting down potentially dangerous reactions within lithium-ion batteries - here 'with..batteries' is a prepositional modifier
This sentence would never be the correct answer on the GMAT. It is unclear what exactly is with...batteries. If we can't tell what a modifier is modifying, the answer choice should be eliminated.
To function as an adjective, a prepositional modifier must be describing not an action but a noun:
The tree by the river provides comfort.
What kind of tree? The tree by the river. By the river is an adjective modifying the noun tree.
The tree provides comfort by blocking the bright sun.
How does the tree provide comfort? By blocking the bright sun. By blocking is an adverb modifying the verb provides.
Hope this helps!
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