Exam 4: A: Number Theory and Cryptography

What is the shared key if Alice and Bob use the Diﬃe-Hellman key exchange protocol with the prime p = 431, the primitive root a = 79 of p = 431, with Alice choosing the secret integer k₁ = 236 and Bob choosing the secret integer k₂ = 334?

(Short Answer)

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Question 141

Encrypt the message "meet me at noon" using the function f(x) = (9x + 1) mod 26.

(Essay)

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Question 142

Use the Euclidean Algorithm to ﬁnd gcd(400, 0).

(Short Answer)

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Question 143

Prove or disprove: The sum of two irrational numbers is irrational.

(True/False)

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Question 144

Find an inverse of 17 modulo 19.

(Short Answer)

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Question 145

ﬁnd the sum and product of each of these pairs of numbers. Express your answer as a base 3 expansion. - $( 202 ) _ { 3 , } ( 122 ) _ { 3 }$

(Short Answer)

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Question 146

Prove or disprove: $\text { If } a \text { and } b \text { are rational numbers (not equal to zero), then } a ^ { b } \text { is rational. }$

(True/False)

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Question 147

Take any three-digit integer, reverse its digits, and subtract. For example, 742 − 247 = 495. The diﬀerence is divisible by 9. Prove that this must happen for all three-digit numbers abc.

(Short Answer)

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Question 148

Convert (2AC)₁₆ to base 10 .

(Short Answer)

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Question 149

What is the shared key if Alice and Bob use the Diﬃe-Hellman key exchange protocol with the prime p = 431, the primitive root a = 79 of p = 431, with Alice choosing the secret integer k₁ = 236 and Bob choosing the secret integer k₂ = 334?

Encrypt the message "meet me at noon" using the function f(x) = (9x + 1) mod 26.

Use the Euclidean Algorithm to ﬁnd gcd(400, 0).

Prove or disprove: The sum of two irrational numbers is irrational.

Find an inverse of 17 modulo 19.

ﬁnd the sum and product of each of these pairs of numbers. Express your answer as a base 3 expansion. - $( 202 ) _ { 3 , } ( 122 ) _ { 3 }$

Prove or disprove: $\text { If } a \text { and } b \text { are rational numbers (not equal to zero), then } a ^ { b } \text { is rational. }$

Take any three-digit integer, reverse its digits, and subtract. For example, 742 − 247 = 495. The diﬀerence is divisible by 9. Prove that this must happen for all three-digit numbers abc.

Convert (2AC)₁₆ to base 10 .

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

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Exam 4: A: Number Theory and Cryptography

What is the shared key if Alice and Bob use the Diﬃe-Hellman key exchange protocol with the prime p = 431, the primitive root a = 79 of p = 431, with Alice choosing the secret integer k1 = 236 and Bob choosing the secret integer k2 = 334?

Encrypt the message "meet me at noon" using the function f(x) = (9x + 1) mod 26.

Use the Euclidean Algorithm to ﬁnd gcd(400, 0).

Prove or disprove: The sum of two irrational numbers is irrational.

Find an inverse of 17 modulo 19.

ﬁnd the sum and product of each of these pairs of numbers. Express your answer as a base 3 expansion. - (202)3,(122)3( 202 ) _ { 3 , } ( 122 ) _ { 3 }(202)3,​(122)3​

Prove or disprove: If a and b are rational numbers (not equal to zero), then ab is rational. \text { If } a \text { and } b \text { are rational numbers (not equal to zero), then } a ^ { b } \text { is rational. } If a and b are rational numbers (not equal to zero), then ab is rational.

Take any three-digit integer, reverse its digits, and subtract. For example, 742 − 247 = 495. The diﬀerence is divisible by 9. Prove that this must happen for all three-digit numbers abc.

Convert (2AC)16 to base 10 .

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

What is the shared key if Alice and Bob use the Diﬃe-Hellman key exchange protocol with the prime p = 431, the primitive root a = 79 of p = 431, with Alice choosing the secret integer k₁ = 236 and Bob choosing the secret integer k₂ = 334?

ﬁnd the sum and product of each of these pairs of numbers. Express your answer as a base 3 expansion. - $( 202 ) _ { 3 , } ( 122 ) _ { 3 }$

Prove or disprove: $\text { If } a \text { and } b \text { are rational numbers (not equal to zero), then } a ^ { b } \text { is rational. }$

Convert (2AC)₁₆ to base 10 .