Find Alice and Bob's common key K by using the Diffie-Hellman-Merkle key exchange scheme with the given values of M, n, a, and b. -$\begin{array}{llll} \hline \mathrm{M} & \mathrm{n} & \mathrm{a} & \mathrm{b} \\ \hline 13 & 11 & 7 & 4 \\ \hline \end{array}$

A) 3 B) 11 C) 5 D) 4 A) 3 B) 11 C) 5 D) 4

$\begin{array} { l } 1 ^ { 2 } + 1 ^ { 2 } = 2 \\ 1 ^ { 2 } + 2 ^ { 2 } = 5 \\ 2 ^ { 2 } + 3 ^ { 2 } = 13 \end{array}$

A) $3 ^ { 2 } + 5 ^ { 2 } = 34$ B) $3 ^ { 2 } + 8 ^ { 2 } = 73$ C) $3 ^ { 2 } + 5 ^ { 2 } = 35$ D) $2 ^ { 2 } + 5 ^ { 2 } = 29$ A) $3 ^ { 2 } + 5 ^ { 2 } = 34$ B) $3 ^ { 2 } + 8 ^ { 2 } = 73$ C) $3 ^ { 2 } + 5 ^ { 2 } = 35$ D) $2 ^ { 2 } + 5 ^ { 2 } = 29$

Exam 5: Number Theory

Solve the problem. -Factorial primes are those of the form n! ± 1 for natural numbers n. Determine the two numbers that are obtained when this formula is applied to the value n = 6, and state whether both, neither, or exactly one of them is prime.

(Essay)

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Question 121

Determine whether the statement is true or false. -The greatest common factor of two relatively prime numbers is always 1.

(True/False)

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Question 122

Use divisibility tests to decide whether the first number is divisible by the second. -615,849; 9

(Multiple Choice)

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Question 123

Determine whether the statement is true or false. -For any natural number n, if you multiply the nth term of the Fibonacci sequence and the nth term of the Lucas sequence, you will obtain the (2n)th term of the Fibonacci sequence.

(True/False)

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Question 124

Solve the problem relating to the Fibonacci sequence. -List the first seven terms of the Fibonacci sequence.

(Multiple Choice)

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Question 125

Use divisibility tests to decide whether the first number is divisible by the second. -938,772; 9

(Multiple Choice)

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Question 126

Give the prime factorization of the number. Use exponents when possible. -2937

(Multiple Choice)

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Question 127

Use divisibility tests to decide whether the first number is divisible by the second. -467,661; 3

(Multiple Choice)

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Question 128

Determine whether or not one or more pairs of twin primes exist between the pair of numbers given. If so, identify the twin primes. -15 and 24

(Multiple Choice)

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Question 129

Determine all values for the digit x that make the first number divisible by the second number. If none exist, so state. -4x21 is divisible by 4.

(Essay)

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Question 130

Find Alice and Bob's common key K by using the Diffie-Hellman-Merkle key exchange scheme with the given values of M, n, a, and b. - 13 11 7 4

(Multiple Choice)

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Question 131

Find the number of divisors of the number. -54

(Multiple Choice)

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Question 132

Find all natural number factors of the number. -110

(Multiple Choice)

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Question 133

Determine whether the statement is true or false. - $\text { If } \mathrm { n } \text { is a composite number, then the Mersenne number } 2 ^ { \mathrm { n } } - 1 \text { is also a composite number. }$

(True/False)

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Question 134

+=2 +=5 +=13

(Multiple Choice)

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Question 135

Use inductive reasoning to find the next equation in the pattern. -1 + 1 = 2 1 + 1 + 2 = 4 1 + 1 + 2 + 3 = 7 1 + 1 + 2 + 3 + 5 = 12

(Multiple Choice)

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Question 136

Determine whether the number is abundant or deficient. -30

(Multiple Choice)

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Question 137

Use divisibility tests to decide whether the first number is divisible by the second. -574,085; 4

(Multiple Choice)

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Question 138

Determine whether the statement is true or false. -For any odd natural number n: If you square the nth term of the Lucas sequence and add 2 you will obtain the (2n)th term of the Lucas sequence.

(True/False)

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Question 139

Determine all values for the digit x that make the first number divisible by the second number. If none exist, so state. -43x6 is divisible by 4.

(Short Answer)

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Question 140

Solve the problem. -Factorial primes are those of the form n! ± 1 for natural numbers n. Determine the two numbers that are obtained when this formula is applied to the value n = 6, and state whether both, neither, or exactly one of them is prime.

Determine whether the statement is true or false. -The greatest common factor of two relatively prime numbers is always 1.

Use divisibility tests to decide whether the first number is divisible by the second. -615,849; 9

Determine whether the statement is true or false. -For any natural number n, if you multiply the nth term of the Fibonacci sequence and the nth term of the Lucas sequence, you will obtain the (2n)th term of the Fibonacci sequence.

Solve the problem relating to the Fibonacci sequence. -List the first seven terms of the Fibonacci sequence.

Use divisibility tests to decide whether the first number is divisible by the second. -938,772; 9

Give the prime factorization of the number. Use exponents when possible. -2937

Use divisibility tests to decide whether the first number is divisible by the second. -467,661; 3

Determine whether or not one or more pairs of twin primes exist between the pair of numbers given. If so, identify the twin primes. -15 and 24

Determine all values for the digit x that make the first number divisible by the second number. If none exist, so state. -4x21 is divisible by 4.

Find Alice and Bob's common key K by using the Diffie-Hellman-Merkle key exchange scheme with the given values of M, n, a, and b. - 13 11 7 4

Find the number of divisors of the number. -54

Find all natural number factors of the number. -110

Determine whether the statement is true or false. - $\text { If } \mathrm { n } \text { is a composite number, then the Mersenne number } 2 ^ { \mathrm { n } } - 1 \text { is also a composite number. }$

+=2 +=5 +=13

Use inductive reasoning to find the next equation in the pattern. -1 + 1 = 2 1 + 1 + 2 = 4 1 + 1 + 2 + 3 = 7 1 + 1 + 2 + 3 + 5 = 12

Determine whether the number is abundant or deficient. -30

Use divisibility tests to decide whether the first number is divisible by the second. -574,085; 4

Determine whether the statement is true or false. -For any odd natural number n: If you square the nth term of the Lucas sequence and add 2 you will obtain the (2n)th term of the Lucas sequence.

Determine all values for the digit x that make the first number divisible by the second number. If none exist, so state. -43x6 is divisible by 4.

The Art of Problem Solving

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Introduction to Logic

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The Real Numbers and Their Representations

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Graphs, Functions, and Systems of Equations and Inequalities

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