Using the Extended Euclidean algorithm,

A$)$ Find the greatest common divisor of $19$ and $999,$ that is$,gcd(999,19).$ Show your work clearly step by step.

B$)$ Express the $gcd(999,19)$ as a linear combination of $999$ and $19.$

C$)$ Compute the multiplicative inverse of $19$ mod $999,$ which is a number between $0$ and $998.$

D$)$ Compute the multiplicative inverse of $999$ mod $19,$ which is a number between $0$ and $18.$

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