Question 1

Explain why f(x) = (2x + 3) mod 26 would not be a good coding function.

Accepted Answer

The answer of Explain why f(x) = (2x + 3)...

Question 2

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
-$$c \equiv 2 a + 4 b ( \bmod 7 )$$

Accepted Answer

The answer of suppose that a and b are integers,...

Question 3

Use the Euclidean algorithm to ﬁnd gcd(34, 21).

Accepted Answer

The answer of Use the Euclidean algorithm to ﬁnd gcd(34,...

Question 4

Find the hexadecimal expansion of (ABC)16+(2F5)16.

Accepted Answer

The answer of Find the hexadecimal expansion of (ABC)16+(2F5)16....

Question 5

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.
-Suppose the ﬁrst digit of the student id X923 4562 is illegible (indicated by X). Can you tell what the ﬁrst digit has to be?

Accepted Answer

The answer of refer to an 8-digit student id at...

Question 6

Use the Euclidean Algorithm to ﬁnd gcd(390, 72).

Accepted Answer

The answer of Use the Euclidean Algorithm to ﬁnd gcd(390,...

Question 7

Find the prime factorization of 45,617.

Accepted Answer

The answer of Find the prime factorization of 45,617....

Question 8

Find an inverse of 6 modulo 7.

Accepted Answer

The answer of Find an inverse of 6 modulo 7....

Question 9

Find the smallest integer $$a > 1 \text { Such that } a + 1 \equiv 2 a ( \bmod 11 ) \text {. }$$

Accepted Answer

The answer of Find the smallest integer \[a > 1...

Question 10

determine whether each of the following "theorems" is true or false. Assume that a, b, c, d,
and m are integers with m > 1.
-$$\text { If } a \equiv b ( \bmod m ) , \text { then } 2 a \equiv 2 b ( \bmod 2 m )$$

Accepted Answer

A) True 
 B)False

Question 11

Convert  (8091)10  to base 2 .

Accepted Answer

Question 12

What is the shared key if Alice and Bob use the Diﬃe-Hellman key exchange protocol with the prime p = 67, the primitive root a = 7 of p = 67, with Alice choosing the secret integer k1 = 12 and Bob choosing the secret integer k2 = 25?

Accepted Answer

Question 13

Find integers a and b such that $$a + b \equiv a - b ( \bmod 5 )$$

Accepted Answer

The answer of Find integers a and b such that...

Question 14

Encode the message "stop at noon" using the function f(x) = (x + 6) mod 26.

Accepted Answer

The answer of Encode the message "stop at noon" using...

Question 15

The cipher text LTDTLLWW was produced by encrypting a plaintext message using the Vigen`ere cipher with the key TEST. What is the plaintext message?

Accepted Answer

The answer of The cipher text LTDTLLWW was produced by...

Question 16

Prove or disprove: For all integers $$a , b , c \text {, if } a \mid b \text { and } b \mid c \text { then } a \mid c$$

Accepted Answer

A) True 
 B)False

Question 17

determine whether each of the following "theorems" is true or false. Assume that a, b, c, d,
and m are integers with m > 1.
-$$\text { If } a \equiv b \left( \bmod m ^ { 2 } \right) , \text { then } a \equiv b ( \bmod m )$$

Accepted Answer

A) True 
 B)False

Question 18

Use the Euclidean algorithm to ﬁnd gcd(300, 700).

Accepted Answer

The answer of Use the Euclidean algorithm to ﬁnd gcd(300,...

Question 19

Find the ﬁrst ﬁve terms of the sequence of four-digit pseudorandom numbers generated by the middle square method starting with 1357.

Accepted Answer

1357, 8414

Question 20

ﬁnd each of these values
-$$\left( 5 ^ { 4 } \bmod 7 \right) ^ { 3 } \bmod 13$$

Accepted Answer

The answer of ﬁnd each of these values
-\[\left( 5 ^...

Exam 4: A: Number Theory and Cryptography

Explain why f(x) = (2x + 3) mod 26 would not be a good coding function.

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≡2a+4b( mod 7)c \equiv 2 a + 4 b ( \bmod 7 )c≡2a+4b(mod7)

Use the Euclidean algorithm to ﬁnd gcd(34, 21).

Find the hexadecimal expansion of (ABC)16+(2F5)16.

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. -Suppose the ﬁrst digit of the student id X923 4562 is illegible (indicated by X). Can you tell what the ﬁrst digit has to be?

Use the Euclidean Algorithm to ﬁnd gcd(390, 72).

Find the prime factorization of 45,617.

Find an inverse of 6 modulo 7.

Find the smallest integer a>1 Such that a+1≡2a( mod 11). a > 1 \text { Such that } a + 1 \equiv 2 a ( \bmod 11 ) \text {. }a>1 Such that a+1≡2a(mod11).

determine whether each of the following "theorems" is true or false. Assume that a, b, c, d, and m are integers with m > 1. - If a≡b( mod m), then 2a≡2b( mod 2m)\text { If } a \equiv b ( \bmod m ) , \text { then } 2 a \equiv 2 b ( \bmod 2 m ) If a≡b(modm), then 2a≡2b(mod2m)

Convert (8091)10 to base 2 .

What is the shared key if Alice and Bob use the Diﬃe-Hellman key exchange protocol with the prime p = 67, the primitive root a = 7 of p = 67, with Alice choosing the secret integer k1 = 12 and Bob choosing the secret integer k2 = 25?

Find integers a and b such that a+b≡a−b( mod 5)a + b \equiv a - b ( \bmod 5 )a+b≡a−b(mod5)

Encode the message "stop at noon" using the function f(x) = (x + 6) mod 26.

The cipher text LTDTLLWW was produced by encrypting a plaintext message using the Vigen`ere cipher with the key TEST. What is the plaintext message?

Prove or disprove: For all integers a,b,c, if a∣b and b∣c then a∣ca , b , c \text {, if } a \mid b \text { and } b \mid c \text { then } a \mid ca,b,c, if a∣b and b∣c then a∣c

determine whether each of the following "theorems" is true or false. Assume that a, b, c, d, and m are integers with m > 1. - If a≡b( mod m2), then a≡b( mod m)\text { If } a \equiv b \left( \bmod m ^ { 2 } \right) , \text { then } a \equiv b ( \bmod m ) If a≡b(modm2), then a≡b(modm)

Use the Euclidean algorithm to ﬁnd gcd(300, 700).

Find the ﬁrst ﬁve terms of the sequence of four-digit pseudorandom numbers generated by the middle square method starting with 1357.

ﬁnd each of these values - (54 mod 7)3 mod 13\left( 5 ^ { 4 } \bmod 7 \right) ^ { 3 } \bmod 13(54mod7)3mod13

The Foundations: Logic and Proofs

A: the Foundations: Logic and Proofs

Basic Structures: Sets, Functions, Sequences, Sums, Matrices

A: Basic Structures: Sets, Functions, Sequences, Sums, Matrices

Algorithms

A: Algorithms

Number Theory and Cryptography

Induction and Recursion

A: Induction and Recursion

Counting

A: Counting

Discrete Probability

A: Discrete Probability

Advanced Counting Techniques

A: Advanced Counting Techniques

Relations

A: Relations

Graphs

A: Graphs

Trees

A: Trees

Boolean Algebra

A: Boolean Algebra

Modeling Computation

A: Modeling Computation

Mathematics Problem Set: Set Theory, Number Theory, Combinatorics, and Boolean Algebra

Filters

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 - $c \equiv 2 a + 4 b ( \bmod 7 )$

Find the smallest integer $a > 1 \text { Such that } a + 1 \equiv 2 a ( \bmod 11 ) \text {. }$

determine whether each of the following "theorems" is true or false. Assume that a, b, c, d, and m are integers with m > 1. - $\text { If } a \equiv b ( \bmod m ) , \text { then } 2 a \equiv 2 b ( \bmod 2 m )$

Convert (8091)₁₀ to base 2 .

What is the shared key if Alice and Bob use the Diﬃe-Hellman key exchange protocol with the prime p = 67, the primitive root a = 7 of p = 67, with Alice choosing the secret integer k₁ = 12 and Bob choosing the secret integer k₂ = 25?

Find integers a and b such that $a + b \equiv a - b ( \bmod 5 )$

Prove or disprove: For all integers $a , b , c \text {, if } a \mid b \text { and } b \mid c \text { then } a \mid c$

determine whether each of the following "theorems" is true or false. Assume that a, b, c, d, and m are integers with m > 1. - $\text { If } a \equiv b \left( \bmod m ^ { 2 } \right) , \text { then } a \equiv b ( \bmod m )$

ﬁnd each of these values - $\left( 5 ^ { 4 } \bmod 7 \right) ^ { 3 } \bmod 13$