Question 1

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 2

Prove or disprove that there are six consecutive composite integers.

Accepted Answer

We can give a constructive pro...

Question 3

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 4

Find the prime factorization of 45617.

Accepted Answer

We see that neither <img  loading="lazy" src="/assets/images/blured-textbook.svg" class="d-inline m-1" alt="blured image" width="139" height="30">, nor 7 divides 45,6...

Question 5

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 6

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

Accepted Answer

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

Question 7

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 8

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 9

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 10

Find the prime factorization of 111111.

Accepted Answer

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

Exam 4: Number Theory and Cryptography

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

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

Prove or disprove that there are six consecutive composite integers.

Correct Answer
verified
We can give a constructive pro...
View Answer

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

Find the prime factorization of 45617.

Correct Answer
verified
We see that neither , nor 7 divides 45,6...
View Answer

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

Correct Answer
verified
Working from right to left in base 16, w...
View Answer

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

Correct Answer
verified

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

Correct Answer
verified

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

Correct Answer
verified
(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.

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

Correct Answer
verified
(a) We see that 302 = 1 · 201 + 101, 201...
View Answer

Find the prime factorization of 111111.

Correct Answer
verified
We see that 2 does not divide 111,111, b...
View Answer

Exam 4: Number Theory and Cryptography

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

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

Prove or disprove that there are six consecutive composite integers.

Correct AnswerverifiedWe can give a constructive pro...View AnswerShow Answer

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) .

Find the prime factorization of 45617.

Correct AnswerverifiedWe see that neither , nor 7 divides 45,6...View AnswerShow Answer

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​?

Correct AnswerverifiedWorking from right to left in base 16, w...View AnswerShow Answer

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

Correct AnswerverifiedShow Answer

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

Correct AnswerverifiedShow Answer

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

Correct Answerverified(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.

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) .

Correct Answerverified(a) We see that 302 = 1 · 201 + 101, 201...View AnswerShow Answer

Find the prime factorization of 111111.

Correct AnswerverifiedWe see that 2 does not divide 111,111, b...View AnswerShow Answer

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

Correct Answer
verified
We can give a constructive pro...
View Answer

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

Correct Answer
verified
We see that neither , nor 7 divides 45,6...
View Answer

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

Correct Answer
verified
Working from right to left in base 16, w...
View Answer

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

Correct Answer
verified

Correct Answer
verified

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

Correct Answer
verified
(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.

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

Correct Answer
verified
(a) We see that 302 = 1 · 201 + 101, 201...
View Answer

Correct Answer
verified
We see that 2 does not divide 111,111, b...
View Answer