Hi Alex,
you're right, both are correct.
"de silentio" or "e silentio".
Anyway, we usually put "de" before words beginning with a consonant and "ex" before words beginning with a vowel.
ex aqua, ex auro, ex argento but
de marmore, de silentio, etc.
That's just the most common rule for using the prepositions. It looks smarter and this is not my opinion, but ancient linguistic tradition.
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