Exam 4: A: Number Theory and Cryptography

Find the prime factorization of 111,111.

Prove or disprove: $\text { If } p \text { and } q \text { are primes } ( > 2 ) \text {, then } p q + 1 \text { is never prime }$

List all positive integers less than 30 that are relatively prime to 20.

Prove or disprove: If f(n)=n²-n+17, then f(n) is prime for all positive integers n.

Find an integer a such that $a \equiv 3 a ( \bmod 7 )$

Encrypt the message BALL using the RSA system with n = 37 · 73 and e = 7, translating each letter into integers (A = 00, B=01, . . . ) and grouping together pairs of integers.

What does a 60-second stop watch read 54 seconds before it reads 19 seconds?

Convert (11101)₂ to base 16 .

Given that gcd(620, 140) = 20, write 20 as a linear combination of 620 and 140.

find each of these values -(123 mod 19 + 342 mod 19) mod 19

Find the sequence of pseudorandom numbers generated by the power generator $x _ { n + 1 } = x _ { n } ^ { 2 }$ seed $x _ { 0 } = 5$

Prove: if $n$ an integer that is not a multiple of 4, then $n ^ { 2 } \equiv 0$ mod 4 or $n ^ { 2 } \equiv 1$ mod 4 .

Find 50! mod 49!.

Express gcd(450, 120) as a linear combination of 120 and 450.

refer to an 8-digit student id at a large university. The eighth digit is a check digit equal to the sum of the first seven digits modulo 7. -Find the check digit of the student id starting with 2365 415.

Use the Euclidean algorithm to find gcd(144, 233).

find the sum and product of each of these pairs of numbers. Express your answer as an octal expansion. - (4274)₈ ,(5366)₈

Find an inverse of 2 modulo 31.

Either find an integer $x$ such that $x \equiv 2$ (mod 6) and $x \equiv 3$ (mod 9) are both true, or else prove that there is no such integer.

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 - $c \equiv 3 a ( \bmod 7 )$