Is it possible to have a 128 bytes RSA public exponent?

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I am trying to decrypt a friend challenge!

He gaves me a text file that contains a public key and an encrypted message. The public key contains a modulo of 128 bytes and an exponent of 128 bytes too.

So I am wondering if this is a public key or a private key because private keys are generally composed of an exponent of the same size than modulo...

Any clue?

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