# Solve Two Mathematical Questions Showing the Steps Clearly

**Question description**

1. Here is a message to you in which I am using RSA system with pulic key n = 2773 and encryption key e = 157.

0245 2040 1698 1439 1364 1758 0946 0881

1979 1130

I have broken my original message into pairs of characters and converted these pairs to numbers as we did in the text. For example, the word "MATH" would be broken into "MA" and "TH" that would be converted to the numbers 1301 and 2008, respectively, and encrypted. Figure out the decryption key d, decrypt the message and answer the question that it asks.

RSA Cryptosystem table.

A | B | C | D | E | F | G | H | I | J | K | L | M |

01 | 02 | 03 | 04 | 05 | 06 | 07 | 08 | 09 | 10 | 11 | 12 | 13 |

N | O | P | Q | R | S | T | U | V | W | X | Y | Z |

14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 |

Euler's theorem

C(pq) = (p-1)(q-1)

n = pq = 2773*157 = 435,361

C(n) = c(pq) = (p-1)(q-1) = 2772*156 = 432,432

encryption E(M) = M^e mod n

M^157 mod 435,361

decryption D(N) = N^d mod n

N^d mod 435,361

d = e^-1 mod c(n)

2. Use mathematical induction to prove that for every n >= 1, if a set has n elements, then

its power set has 2^n elements.

Must Show all Works and proper steps and signs. these two questions are seperated and not related to each other.

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