Exam 4: A: Number Theory and Cryptography

Prove or disprove: The sum of two primes is a prime.

Find the two's complement of −13.

Use the Vigen`ere cipher with key NOW to encrypt the message SUMMER.

Use the Euclidean algorithm to find gcd(44, 52).

Prove: if $n$ is an integer that is not a multiple of 3 , then $n ^ { 2 } \equiv 1$ mod 3 .

Convert (6253)₈ to base 2 .

Find an inverse of 5 modulo 12.

Alice has the public key (n, e) = (2623, 13) with corresponding private key d = 1357, and she wants to send the message LAST CALL to her friends so that they know she sent it. What should she send to her friends, assuming she signs the message using the RSA cryptosystem?

Find an inverse of 5 modulo 17.

Find lcm $( 20 ! , 12 ! )$ by directly finding the smallest positive multiple of both numbers.

Prove or disprove: A positive integer congruent to 1 modulo 4 cannot have a prime factor congruent to 3 modulo 4.

Prove or disprove: If p and q are primes $( > 2 )$ then p + q is composite.

suppose that a and b are integers, a ≡ 4 (mod 7), and b ≡ 6 (mod 7). Find the integer c with 0 ≤ c ≤ 6 such that - $c \equiv a ^ { 2 } - b ^ { 2 } ( \bmod 7 )$

Find lcm $\left( 2 ^ { 89 } , 2 ^ { 346 } \right)$ by directly finding the smallest positive multiple of both numbers.

Solve the linear congruence 31x ≡ 57 (mod 61).

Find the prime factorization of 8,827.

Find the prime factorization of 510,510.

find each of these values - $\left( 12 ^ { 2 } \bmod 17 \right) ^ { 3 } \bmod 11$

Use Fermat's little theorem to find $9 ^ { 45 }$ mod 23 .

Suppose that the lcm of two numbers is 400 and their gcd is 10. If one of the numbers is 50, find the other number.