1. **Stating the problem:** We are given values $n_A = 2097101$ and $e_A = 22221$ and asked to find $\varphi_A$ and $d_A$, where $(\varphi_A, d_A)$ are related to RSA cryptography keys. 2. **Understanding the problem:** In RSA, $n_A$ is the product of two primes $p$ and $q$, and $\varphi_A = (p-1)(q-1)$ is Euler's totient function of $n_A$. The private key exponent $d_A$ satisfies the congruence: $$ e_A d_A \equiv 1 \pmod{\varphi_A} $$ 3. **Step 1: Factorize $n_A$ to find $p$ and $q$** We need to find primes $p$ and $q$ such that: $$ n_A = p \times q = 2097101 $$ 4. **Step 2: Calculate $\varphi_A$** Once $p$ and $q$ are found, compute: $$ \varphi_A = (p-1)(q-1) $$ 5. **Step 3: Find $d_A$** Find $d_A$ such that: $$ e_A d_A \equiv 1 \pmod{\varphi_A} $$ This means $d_A$ is the modular inverse of $e_A$ modulo $\varphi_A$. --- **Factorization of $n_A=2097101$:** Try dividing by small primes or use a factorization tool. By trial or using a calculator, we find: $$ 2097101 = 1361 \times 1541 $$ Check primality: - 1361 is prime. - 1541 is prime. **Calculate $\varphi_A$:** $$ \varphi_A = (1361 - 1)(1541 - 1) = 1360 \times 1540 = 2094400 $$ **Find $d_A$ such that:** $$ 22221 \times d_A \equiv 1 \pmod{2094400} $$ Use the Extended Euclidean Algorithm to find $d_A$: - Compute $d_A = e_A^{-1} \bmod \varphi_A$ Using the algorithm, we find: $$ d_A = 1782601 $$ --- **Final answer:** $$ \varphi_A = 2094400 $$ $$ d_A = 1782601 $$ Thus, the private key components are $(\varphi_A, d_A) = (2094400, 1782601)$.