my p is 23 my q is 29 and my e is 67 find my d

To find the value of \( d \), you need to compute the modular multiplicative inverse of \( e \) modulo \( \phi(n) \), where \( n = p \cdot q \) and \( \phi(n) = (p-1)(q-1) \).

Given:
- \( p = 23 \)
- \( q = 29 \)
- \( e = 67 \)

First, calculate \( n \) and \( \phi(n) \):

\[ n = p \cdot q = 23 \cdot 29 = 667 \]

\[ \phi(n) = (p-1)(q-1) = (23-1)(29-1) = 22 \cdot 28 = 616 \]

Now, we need to find \( d \) such that:

\[ e \cdot d \equiv 1 \pmod{\phi(n)} \]

or

\[ 67 \cdot d \equiv 1 \pmod{616} \]

We can use the Extended Euclidean Algorithm to find \( d \).

1. Apply the Euclidean Algorithm to find the gcd of 67 and 616:
   \[
   \begin{align*}
   616 &= 67 \cdot 9 + 13 \\
   67 &= 13 \cdot 5 + 2 \\
   13 &= 2 \cdot 

Learn how to find the modular inverse d using the Extended Euclidean Algorithm with p = 23, q = 29, and e = 67. This problem involves modular arithmetic and Euler's Theorem, essential concepts in number theory.

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