Question 1

Use the Euclidean algorithm to find
(a) $\operatorname { gcd } ( 203,101 )$.
(b) $\operatorname { gcd } ( 34,21 )$.

Accepted Answer

(a) We have $203 = 2 \cdot 101 + 1$ and $101 = 101 \cdot 1$. It follows that $\operatorname { gcd } ( 203,101 ) = 1$.
(b) We have $34 = 1 \cdot 21 + 13,21 = 1 \cdot 13 + 8,13 = 1 \cdot 8 + 5,8 = 1 \cdot 5 + 3,5 = 1 \cdot 3 + 2,3 = 1 \cdot 2 + 1$, $2 = 2 \cdot 1$ Hence $\operatorname { gcd } ( 34,21 ) = 1$.

Question 2

Prove or disprove that a positive integer congruent to 1 modulo 4 cannot have a prime factor congruent to

Accepted Answer

This is false, since 9 = 4 · 2 + 1 = 3 · 3.

Question 3

Find each of the following values.
(a) $18 \bmod 7$
(b) $- 88 \bmod 13$
(c) $289 \bmod 17$

Accepted Answer

(a) We have 18 = 2 · 7 + 4. Hence 18 mod 7 = 4. (b) We have −88 = −7 · 13 + 3. Hence −88 mod 13 = 3. (c) We have 289 = 17 · 17. Hence 289 mod 17 = 0.

Question 4

What is the hexadecimal expansion of the $$( \mathrm { ABC } ) _ { 16 } + ( 2 \mathrm {~F} 5 ) _ { 16 } ?$$

Accepted Answer

Working from right to left in base 16, w

Question 5

The binary expansion of an integer is (110101)2. What is the base 10 expansion of this integer?

Accepted Answer

The answer of The binary expansion of an integer is...

Question 6

Use the Euclidean algorithm to find
(a) $\operatorname { gcd } ( 201,302 )$.
(b) $\operatorname { gcd } ( 144,233 )$.

Accepted Answer

(a) We see that 302 = 1 · 201 + 101, 201

Question 7

$$\text { Decide whether } 175 \equiv 22 ( \bmod 17 )$$

Accepted Answer

The answer of \[\text { Decide whether } 175 \equiv...

Question 8

Find the prime factorization of 45617.

Accepted Answer

We see that neither @#LAT-DLM&, nor 7 divides 45,6

Question 9

Prove or disprove that there are six consecutive composite integers.

Accepted Answer

We can give a constructive pro

Question 10

Find the prime factorization of 111111.

Accepted Answer

We see that 2 does not divide 111,111, b

Use the Euclidean algorithm to find (a) $\operatorname { gcd } ( 203,101 )$ . (b) $\operatorname { gcd } ( 34,21 )$ .

Prove or disprove that a positive integer congruent to 1 modulo 4 cannot have a prime factor congruent to

Find each of the following values. (a) $18 \bmod 7$ (b) $- 88 \bmod 13$ (c) $289 \bmod 17$

What is the hexadecimal expansion of the $( \mathrm { ABC } ) _ { 16 } + ( 2 \mathrm {~F} 5 ) _ { 16 } ?$

The binary expansion of an integer is (110101)2. What is the base 10 expansion of this integer?

Use the Euclidean algorithm to find (a) $\operatorname { gcd } ( 201,302 )$ . (b) $\operatorname { gcd } ( 144,233 )$ .

$\text { Decide whether } 175 \equiv 22 ( \bmod 17 )$

Find the prime factorization of 45617.

Prove or disprove that there are six consecutive composite integers.

Find the prime factorization of 111111.

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

A: 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

Exam 4: Number Theory and Cryptography

Use the Euclidean algorithm to find (a) gcd⁡(203,101)\operatorname { gcd } ( 203,101 )gcd(203,101) . (b) gcd⁡(34,21)\operatorname { gcd } ( 34,21 )gcd(34,21) .

Prove or disprove that a positive integer congruent to 1 modulo 4 cannot have a prime factor congruent to

Find each of the following values. (a) 18 mod 718 \bmod 718mod7 (b) −88 mod 13- 88 \bmod 13−88mod13 (c) 289 mod 17289 \bmod 17289mod17

What is the hexadecimal expansion of the (ABC)16+(2 F5)16?( \mathrm { ABC } ) _ { 16 } + ( 2 \mathrm {~F} 5 ) _ { 16 } ?(ABC)16​+(2 F5)16​?

The binary expansion of an integer is (110101)2. What is the base 10 expansion of this integer?

Use the Euclidean algorithm to find (a) gcd⁡(201,302)\operatorname { gcd } ( 201,302 )gcd(201,302) . (b) gcd⁡(144,233)\operatorname { gcd } ( 144,233 )gcd(144,233) .

Decide whether 175≡22( mod 17)\text { Decide whether } 175 \equiv 22 ( \bmod 17 ) Decide whether 175≡22(mod17)

Find the prime factorization of 45617.

Prove or disprove that there are six consecutive composite integers.

Find the prime factorization of 111111.

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

A: 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

Use the Euclidean algorithm to find (a) $\operatorname { gcd } ( 203,101 )$ . (b) $\operatorname { gcd } ( 34,21 )$ .

Find each of the following values. (a) $18 \bmod 7$ (b) $- 88 \bmod 13$ (c) $289 \bmod 17$

What is the hexadecimal expansion of the $( \mathrm { ABC } ) _ { 16 } + ( 2 \mathrm {~F} 5 ) _ { 16 } ?$

Use the Euclidean algorithm to find (a) $\operatorname { gcd } ( 201,302 )$ . (b) $\operatorname { gcd } ( 144,233 )$ .

$\text { Decide whether } 175 \equiv 22 ( \bmod 17 )$