To find the inverse of $101 \mod 4620$, we need to determine the integer $x$ such that:

$101 \cdot x \equiv 1 \pmod{4620}$

This is done using the Extended Euclidean Algorithm, which helps us find $x$ and verify that 101 and 4620 are coprime (i.e., their greatest common divisor is 1). Let's go through the steps.

Step 1: Apply the Euclidean Algorithm to find $\gcd(101, 4620)$

We repeatedly apply the division algorithm:

$4620 = 45 \cdot 101 + 75$ $101 = 1 \cdot 75 + 26$ $75 = 2 \cdot 26 + 23$ $26 = 1 \cdot 23 + 3$ $23 = 7 \cdot 3 + 2$ $3 = 1 \cdot 2 + 1$ $2 = 2 \cdot 1 + 0$

Since the greatest common divisor (gcd) is 1, we can proceed.

Step 2: Use the Extended Euclidean Algorithm to express gcd as a linear combination

We now work backward to express 1 as a combination of 101 and 4620:

$1 = 3 - 1 \cdot 2$ Substitute $2 = 23 - 7 \cdot 3$: $1 = 3 - 1 \cdot (23 - 7 \cdot 3) = 8 \cdot 3 - 1 \cdot 23$ Substitute $3 = 26 - 1 \cdot 23$: $1 = 8 \cdot (26 - 1 \cdot 23) - 1 \cdot 23 = 8 \cdot 26 - 9 \cdot 23$ Substitute $23 = 75 - 2 \cdot 26$: $1 = 8 \cdot 26 - 9 \cdot (75 - 2 \cdot 26) = 26 \cdot (8 + 18) - 9 \cdot 75 = 26 \cdot 26 - 9 \cdot 75$ Substitute $26 = 101 - 1 \cdot 75$: $1 = 26 \cdot (101 - 1 \cdot 75) - 9 \cdot 75 = 26 \cdot 101 - 35 \cdot 75$ Substitute $75 = 4620 - 45 \cdot 101$: $1 = 26 \cdot 101 - 35 \cdot (4620 - 45 \cdot 101) = 26 \cdot 101 - 35 \cdot 4620 + 1575 \cdot 101$ $1 = 1601 \cdot 101 - 35 \cdot 4620$

Thus, $x = 1601$.

Step 3: Final Answer

Therefore, the inverse of $101 \mod 4620$ is $\boxed{1601}$.

Would you like a further breakdown of any part of the process? Here are 5 related questions to expand on this:

1. How does the Extended Euclidean Algorithm work in general?
2. What is the importance of modular inverses in cryptography?
3. Can every number have a modular inverse?
4. How is modular arithmetic applied in number theory problems?
5. What happens if two numbers are not coprime in this context?

Tip: Modular inverses only exist if the numbers involved are coprime. This is guaranteed when their greatest common divisor is 1!