Public Key Cryptography: Diffie-Hellman Key Exchange

Please watch this video it's very short. If you already know cryptography and really good at it please start from 3:51.

I didn't understand the step at 4:39, can someone here to explain?

Why is $16^{54} \mod 17$ is equal to

$3 ^ {24\times54} \mod17 \equiv 1$?

He just wrote that. Alice didn't knew bob's private number how did he raised $3 ^{24}$?

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It was pretty disorienting how he completely skipped over the exponentiation explanation –  Richie Frame Jun 27 '14 at 9:51
His newer video explained that part in great detail youtube.com/watch?v=YEBfamv-_do –  Richie Frame Jun 27 '14 at 11:11
Thanks a lot @RichieFrame –  Abhimanyu Aryan Jun 27 '14 at 11:48

Alice didn't know that $16\equiv 3^{24}\mod 17$. The video just made it clearer that they were intrisically computing the same value.
I'm guessing that the point the video was trying to make was that $$3^{54\times24}\equiv3^{24\times54}\equiv 16^{54}\equiv15^{24}\equiv1\mod 17.$$