In a RSA cryptosystem, a participant A uses two prime numbers $p = 13$ and $q = 17$ to generate her public and private keys. If the public key of A is 35, then the private key of A is __________.

In the RSA cryptosystem:

1. Calculate the modulus:

   $$n = p \times q = 13 \times 17 = 221$$

2. Calculate Euler's totient function $\phi(n)$:

   $$\phi(n) = (p - 1)(q - 1) = (13 - 1)(17 - 1) = 12 \times 16 = 192$$

3. The public key exponent is $e = 35$.

4. The private key exponent $d$ is the multiplicative inverse of $e$ modulo $\phi(n)$:

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

   $$35 \cdot d \equiv 1 \pmod{192}$$

Using the Extended Euclidean Algorithm:

Expressing $1$ as a linear combination of $35$ and $192$:

Thus:

Therefore, the private key $d = 11$.