Question 1

Find a div m and a mod m when $$a = - 33 , m = 67$$

Accepted Answer

The answer of Find a div m and a mod...

Question 2

Find the two's complement of 9.

Accepted Answer

The answer of Find the two's complement of 9....

Question 3

Prove or disprove that $$30 \text { ! ends in exactly seven } 0 \text { 's. }$$

Accepted Answer

A) True 
 B)False

Question 4

(a) Find two positive integers, each with exactly three positive integer factors greater than 1. (b) Prove that there are an inﬁnite number of positive integers, each with exactly three positive integer factors greater than 1.

Accepted Answer

(a) 8,27
(b) Any int

Question 5

Suppose that a computer has only the memory locations 0, 1, 2, . . . , 19. Use the hashing function h where h(x) = (x + 5) mod 20 to determine the memory locations in which 57, 32, and 97 are stored.

Accepted Answer

The answer of Suppose that a computer has only the...

Question 6

Convert  (10,000)10  to base  2 .

Accepted Answer

Question 7

ﬁnd the sum and product of each of these pairs of numbers. Express your answer as an octal expansion.
-$( 371 ) _ { 8 }$ $( 624  )_{8}$

Accepted Answer

The answer of ﬁnd the sum and product of each...

Question 8

ﬁnd the sum and product of each of these pairs of numbers. Express your answer as a hexadecimal expansion.
- (E3A)16 , (B5F8)16

Accepted Answer

Question 9

Encrypt the message NEED HELP by translating the letters into numbers (A=0, B=1, . . ., Z=25), applying the encryption function f(p) = (3p + 7) mod 26, and then translating the numbers back into letters.

Accepted Answer

Encrypted

Question 10

Find 18 mod 7.

Accepted Answer

The answer of Find 18 mod 7....

Question 11

Convert  (271)8 to base 2 .

Accepted Answer

Question 12

Express gcd(84, 18) as a linear combination of 18 and 84.

Accepted Answer

The answer of Express gcd(84, 18) as a linear combination...

Question 13

Solve the linear congruence 54x ≡ 12 (mod 73) given that the inverse of 54 modulo 73 is 23.

Accepted Answer

The answer of Solve the linear congruence 54x ≡ 12...

Question 14

Use the Euclidean Algorithm to ﬁnd gcd(900, 140).

Accepted Answer

The answer of Use the Euclidean Algorithm to ﬁnd gcd(900,...

Question 15

Use the Euclidean Algorithm to ﬁnd gcd(128, 729).

Accepted Answer

The answer of Use the Euclidean Algorithm to ﬁnd gcd(128,...

Question 16

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

Accepted Answer

A) True 
 B)False

Question 17

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

Accepted Answer

A) True 
 B)False

Question 18

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

Accepted Answer

A) True 
 B)False

Question 19

Explain in words the difference between $a \mid b$ and $\frac { b } { a }$

Accepted Answer

@#LAT-DLM& is a stat

Question 20

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

Accepted Answer

A) True 
 B)False

Exam 4: A: Number Theory and Cryptography

Find a div m and a mod m when a=−33,m=67a = - 33 , m = 67a=−33,m=67

Find the two's complement of 9.

Prove or disprove that 30 ! ends in exactly seven 0 ’s. 30 \text { ! ends in exactly seven } 0 \text { 's. }30 ! ends in exactly seven 0 ’s.

(a) Find two positive integers, each with exactly three positive integer factors greater than 1. (b) Prove that there are an inﬁnite number of positive integers, each with exactly three positive integer factors greater than 1.

Suppose that a computer has only the memory locations 0, 1, 2, . . . , 19. Use the hashing function h where h(x) = (x + 5) mod 20 to determine the memory locations in which 57, 32, and 97 are stored.

Convert (10,000)10 to base 2 .

ﬁnd the sum and product of each of these pairs of numbers. Express your answer as an octal expansion. - (371)8( 371 ) _ { 8 }(371)8​ (624)8( 624 )_{8}(624)8​

ﬁnd the sum and product of each of these pairs of numbers. Express your answer as a hexadecimal expansion. - (E3A)16 , (B5F8)16

Encrypt the message NEED HELP by translating the letters into numbers (A=0, B=1, . . ., Z=25), applying the encryption function f(p) = (3p + 7) mod 26, and then translating the numbers back into letters.

Find 18 mod 7.

Convert (271)8 to base 2 .

Express gcd(84, 18) as a linear combination of 18 and 84.

Solve the linear congruence 54x ≡ 12 (mod 73) given that the inverse of 54 modulo 73 is 23.

Use the Euclidean Algorithm to ﬁnd gcd(900, 140).

Use the Euclidean Algorithm to ﬁnd gcd(128, 729).

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

Prove or disprove: For all integers a,b,c,d, if a∣b and c∣d, then (a+c)∣(b+d). a , b , c , d \text {, if } a \mid b \text { and } c \mid d \text {, then } ( a + c ) \mid ( b + d ) \text {. }a,b,c,d, if a∣b and c∣d, then (a+c)∣(b+d).

Prove or disprove: For all integers a,b,c,d, if a∣b and c∣d, then (ac)∣(b+d). a , b , c , d \text {, if } a \mid b \text { and } c \mid d \text {, then } ( a c ) \mid ( b + d ) \text {. }a,b,c,d, if a∣b and c∣d, then (ac)∣(b+d).

Explain in words the difference between a∣ba \mid ba∣b and ba\frac { b } { a }ab​

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 a≡b( mod 2m). \text { If } a \equiv b ( \bmod m ) , \text { then } a \equiv b ( \bmod 2 m ) \text {. } If a≡b(modm), then a≡b(mod2m).

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

Find a div m and a mod m when $a = - 33 , m = 67$

Prove or disprove that $30 \text { ! ends in exactly seven } 0 \text { 's. }$

Convert (10,000)₁₀ to base 2 .

ﬁnd the sum and product of each of these pairs of numbers. Express your answer as an octal expansion. - $( 371 ) _ { 8 }$ $( 624 )_{8}$

ﬁnd the sum and product of each of these pairs of numbers. Express your answer as a hexadecimal expansion. - (E3A)₁₆ , (B5F8)₁₆

Convert (271)₈ to base 2 .

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

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

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

Explain in words the difference between $a \mid b$ and $\frac { b } { a }$

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