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!

Learn how to find the inverse of 101 modulo 4620 using the Extended Euclidean Algorithm. This step-by-step solution covers key number theory concepts such as modular arithmetic, gcd, and the importance of coprime numbers. Suitable for Grades 11-12 and undergraduate students.

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