Unit 2 Review

Identify the greatest common divisor of the following pair of integers. 2^2 · 3^3 · 5 · 7 · 11^2 · 13 and 2^9 · 3^8 · 11^1 · 17

2^2 · 3^3 · 11^1

Which equations arise when the steps of the Euclidean algorithm are reversed to express the greatest common divisor of each of these pairs of integers as a linear combination of these integers? (0, 224)

224 = s · 0 + 1 · 224 for some integer s

Convert the binary expansion of each of the following integers to a decimal expansion. The decimal expansion of (1 1001)2 is

25

Find the sum and product of each of these pairs of numbers. Express your answers as base 3 expansions. The sum of the numbers (112)3 and (210)3 is _______ and their product is ____

The sum of the numbers (112)3 and (210)3 is 1. ( 1022 )3 and their product is 2. ( 101220 )3

Find the sum and product of each of these pairs of numbers. Express your answers as base 3 expansions. (120021)3 and (2012)3

The sum of the numbers (120021)3 and (2012)3 is ( 122110 )3 and their product is (1020100022 )3

Identify the sum and product of each of these pairs of numbers. The sum of the numbers (1AE)16 and (BBD)16

The sum of the numbers (1AE)16 and (BBD)16 is ( D6B )16 and their product is ( 13B776)16.

Find the sum and product of each of these pairs of numbers. Express your answers as base 3 expansions. (20001)3 and (1111)3

The sum of the numbers (20001)3 and (1111)3 is ( 21112 )3 and their product is ( 22221111)3

Identify the sum and product of each of these pairs of numbers. The sum of the numbers (20CBA)16 and (A02)16

The sum of the numbers (20CBA)16 and (A02)16 is ( 216BC )16 and their product is (14835D74 )16.

Find the sum and product of each of these pairs of numbers. Express your answers as base 3 expansions. (2112)3 and (12100)3

The sum of the numbers (2112)3 and (12100)3 is ( 21212 )3 and their product is (111102200)

Find the sum and product of each of these pairs of numbers. Express your answers as octal expansions. The sum of the numbers (54321)8 and (3457)8

The sum of the numbers (54321)8 and (3457)8 is ( 60000 )8 and their product is (237402537 )8

Find the sum and product of each of these pairs of numbers. Express your answers as octal expansions. The sum of the numbers (6001)8 and (272)8

The sum of the numbers (6001)8 and (272)8 is ( 6273 )8 and their product is ( 2134272)8.

Find the sum and product of each of these pairs of numbers. Express your answers as octal expansions. The sum of the numbers (763)8 and (144)8

The sum of the numbers (763)8 and (144)8 is ( 1127 )8 and their product is ( 141354 )8.

Identify the sum and product of each of these pairs of numbers. The sum of the numbers (ABCDE)16 and (1111)16

The sum of the numbers (ABCDE)16 and (1111)16 is ( ACDEF )16 and their product is (B74148BE )16.

Identify the sum and product of each of these pairs of numbers. The sum of the numbers (E0000E)16 and (BAAB)16

The sum of the numbers (E0000E)16 and (BAAB)16 is ( E0BAB9 )16 and their product is (A355AA355A )16.

Evaluate these quantities. -101 mod 13 =

-101 mod 13 =

Which of these equations is produced as a step when the Euclidean algorithm is used to find the gcd of given integers? (13671, 1470) (Check all that apply.)

13671 = 9 · 1470 + 441 441 = 3 · 147 + 0 1470 = 3 · 441 + 147

List five integers that are congruent to 4 modulo 10:

4, 24, 44, 64, and 84

Convert the hexadecimal expansion of each of the following integers to a binary expansion. The binary expansion of (80E)16 is

1000 0000 1110

Identify the quotient and remainder when -1 is divided by 2.

(-1,1)

Identify the quotient and remainder when -111 is divided by 11.

(-11,10)

Identify the quotient and remainder when 0 is divided by 19.

(0,0)

Identify the quotient and remainder when 2 is divided by 5.

(0,2)

Identify the quotient and remainder when 19 is divided by 7.

(2,5)

Identify the quotient and remainder when 789 is divided by 23.

(34,7)

Identify the quotient and remainder when 5 is divided by 1.

(5,0)

Identify the quotient and remainder when 1065 is divided by 13.

(81,12)

Which of these equations is produced as a step when the Euclidean algorithm is used to find the gcd of given integers? (101, 100) (Check all that apply.)

101 = 1 · 100 + 1 100 = 100 · 1 + 0

Convert the hexadecimal expansion of each of the following integers to a binary expansion. The binary expansion of (ABBA)16 is

1010 1011 1011 1010

Evaluate these quantities. -21 mod 2=

-21 mod 2= 1

Identify the integers that are congruent to 5 modulo 15. (Check all that apply.)

-25 65

Identify the greatest common divisor of the following pair of integers. 2^2 · 7^1 and 5^3 · 13^1

1

Convert the hexadecimal expansion of each of the following integers to a binary expansion. The binary expansion of (135AA)16 is

1 0011 0101 1010 1010

Convert the decimal expansion of each of the following integers to a binary expansion. 321 The binary expansion of 321 is

1 0100 0001

Convert the decimal expansion of each of the following integers to a binary expansion. 100632 The binary expansion of 100632 is

1 1000 1001 0001 1000

Which equations arise when the steps of the Euclidean algorithm are reversed to express the greatest common divisor of each of these pairs of integers as a linear combination of these integers? (123, 2347) (Check all that apply.)

1 = 10 − 3 · 3 1 = 37 · 10 − 3 · 123 1 = 37 · 2347 − 706 · 123

Which equations arise when the steps of the Euclidean algorithm are reversed to express the greatest common divisor of each of these pairs of integers as a linear combination of these integers? (34, 55) (Check all that apply.)

1 = 2 · 8 - 3 · 5 1 = 13 · 21 - 8 · 34 1 = 5 · 8 - 3 · 13

Which equations arise when the steps of the Euclidean algorithm are reversed to express the greatest common divisor of each of these pairs of integers as a linear combination of these integers? (21, 44) (Check all that apply.)

1 = 21 - 10 · 2 1 = 21 · 21 - 10 · 44

Identify the correct step to prove that if a is an integer other than 0, then 1 divides a.

1 | a since a = 1 • a

Find the prime factorization of each of these integers, and use each factorization to answer the questions posed. The greatest prime factor of 103 is

103

Identify the greatest common divisor of the following pair of integers. The greatest common divisor of 0 and 11 is

11

Convert the decimal expansion of each of the following integers to a binary expansion. 1023 The binary expansion of 1023 is

11 1111 1111

Which of these equations is produced as a step when the Euclidean algorithm is used to find the gcd of given integers? (11, 1)

11 = 11 · 1 + 0

Convert the hexadecimal expansion of each of the following integers to a binary expansion. The binary expansion of (DDFACED)16 is

1101 1101 1111 1010 1100 1110 1101

Which of these equations is produced as a step when the Euclidean algorithm is used to find the gcd of given integers? (1111, 111) (Check all that apply.)

111 = 111 · 1 + 0 1111 = 10 · 111 + 1

Which of these equations is produced as a step when the Euclidean algorithm is used to find the gcd of given integers? (11891, 2238) (Check all that apply.)

11891 = 5 · 2238 + 701 2238 = 3 · 701 + 135 701 = 5 · 135 + 26 26 = 5 · 5 + 1 135 = 5 · 26 + 5 5 = 5 · 1 + 0

Find the sum and product of each of these pairs of numbers. Express your answers as binary expansions. The sum of the numbers (1110 1111)2 and (1011 1110)2 is _______________ and their product is _____________

1. 1 1010 1101 2. 1011 0001 0110 0010

Find the sum and product of each of these pairs of numbers. Express your answers as binary expansions. The sum of the numbers (100 0111)2 and (111 1000)2 is______________ And Their product is _______________-.

1. 1011 1111 2. 10 0001 0100 1000

The statement "if a | bc, where a, b, and c are positive integers and a ≠ 0, then a | b or a | c" is Identify the correct statement to justify your answer.

1. False 2. Using the counterexample 4| (2*2) but 4 does not divide 2, the given statement is false.

Evaluate these quantities. 144 mod 7 =

144 mod 7 = 4

Find the prime factorization of each of these integers, and use each factorization to answer the questions posed. The greatest prime factor of 289 is

17

Identify the greatest common divisor of the following pair of integers. 18 and 18^18

18

Find the least common multiple of each of these pair of integers. 18, 18^18

18^18

Evaluate these quantities. 199 mod 19 =

199 mod 19 = 9

Identify the positive integers that are not relatively prime to 28. (Check all that apply.)

2 7 4 The unique prime factors of 28 are 2 and 7. Therefore, all integers that are divisible by 2 or by 7 are not relatively prime to 28.

Which equations arise when the steps of the Euclidean algorithm are reversed to express the greatest common divisor of each of these pairs of integers as a linear combination of these integers? (3454, 4666) (Check all that apply.)

2 = 58 − 14 · 4 2 = 44 · 120 − 29 · 182 2 = 293 · 3454 − 835 · 1212

Identify the greatest common divisor of the following pair of integers. 2 · 3 · 7 · 11 and 2 · 3 · 7 · 11

2*3*7*11

Which of these equations is produced as a step when the Euclidean algorithm is used to find the gcd of given integers? (123, 277) (Check all that apply.)

277 = 2 · 123 + 31 123 = 3 · 31 + 30 30 = 30 · 1 + 0 31 = 1 · 30 + 1

Find the prime factorization of each of these integers, and use each factorization to answer the questions posed. The smallest prime factor of 899 is

29

Identify the integers that are prime. (Check all that apply.)

29 71 97

Identify the greatest common divisor of the following pair of integers. 2^5 · 3^1 · 5^2 and 2^3 · 3^2 · 5^3

2^3 · 3^1 · 5^2

Find the least common multiple of each of these pair of integers. 2^3 · 3^4 · 5^5 and 2^1 · 3^2 · 5^2

2^3 · 3^4 · 5^5

Identify the least common multiple of two integers if their product is 2^7⋅3^8⋅5^2⋅7^11. and their greatest common divisor is 2^3⋅3^4⋅5

2^4 · 3^4 · 5 · 7^11

Find the least common multiple of each of these pair of integers. 2^5 · 7^3 and 5^4 · 13^2

2^5 · 5^4 · 7^3 · 13^2

Find the least common multiple of each of these pair of integers. 2^4 · 3^1 · 5 · 7 · 11^2 · 13 and 2^8 · 3^4 · 11^1 · 17^14

2^8 · 3^4 · 5 · 7 · 11^2 · 13 · 17^14

Identify the prime factorization of 10!.

2^8 · 3^4 · 5^2 · 7

Which equations arise when the steps of the Euclidean algorithm are reversed to express the greatest common divisor of each of these pairs of integers as a linear combination of these integers? (117, 213) (Check all that apply.)

3 = 1 · 12 − 1 · 9 3 = 2 · 96 − 9 · 21 3 = 11 · 213 − 20 · 117

Convert the binary expansion of each of the following integers to a decimal expansion. The decimal expansion of (111 1100 0001 1111)2 is

31775

Find the prime factorization of each of these integers, and use each factorization to answer the questions posed. The greatest prime factor of 15 is

5

Find the prime factorization of each of these integers, and use each factorization to answer the questions posed. The smallest prime factor of 85 is

5

Identify the numbers that are divisible by 17. (Check all that apply.)

68 357 A number is divisible if the given divisor divides the number with zero remainder.

Convert the binary expansion of each of the following integers to a decimal expansion. The decimal expansion of (10 1011 0101)2 is

693

factorization to answer the questions posed. The greatest prime factor of 2401 is

7

Let m be a positive integer. Show that a ≡ b (mod m) if a mod m = bmod m. Drag the necessary statements and drop them into the appropriate blank to build your proof.

Proof Method: Direct proof Proof's assumption: a mod m= b mod m Conclusion: a= b (mod m)

Convert the binary expansion of each of the following integers to a hexadecimal expansion. The hexadecimal notation of (0111 0111 0111 0111)2 is

7777

Convert the binary expansion of each of the following integers to a decimal expansion. The decimal expansion of (11 0111 1110)2 is

894

Convert the binary expansion of each of the following integers to a hexadecimal expansion. The hexadecimal notation of (1001 1001 1001 1001)2 is

9999

Convert the binary expansion of each of the following integers to a hexadecimal expansion. The hexadecimal notation of (1010 1010 1010)2 is

AAA

Identify the correct steps involved in proving that am + 1 is composite if a and m are integers greater than 1 and m is odd. (Check all that apply.)

As m is odd, we can write am + 1 = (a + 1)(am - 1 - am - 2 + am - 3 - am - 4 + ... + 1). As both a and m are greater than 1, we have 1 < a + 1 < am + 1. Thus, a + 1 is a proper factor of am + 1. Thus, we can express am + 1 as a product of two proper factors; so, am + 1 is composite.

Find the least common multiple of each of these pair of integers. 0 and 5

Does Not Exist

Convert the binary expansion of each of the following integers to a hexadecimal expansion. The hexadecimal notation of (1111 0110)2 is

F6

Find the least common multiple of each of these pair of integers. The least common multiple of 2 · 3 · 5 · 11 and 2 · 3 · 5 · 11 is 1.

FALSE

Find the sum and product of each of these pairs of numbers. Express your answers as binary expansions. The sum of the numbers (10 1010 1010)2 and (1 1111 0010)2 is_____________ and their product is ____________

The sum of the numbers (10 1010 1010)2 and (1 1111 0010)2 is 1. ( 10010011100 )2 and their product is 2. ( 1010010111010110100

Which equations arise when the steps of the Euclidean algorithm are reversed to express the greatest common divisor of each of these pairs of integers as a linear combination of these integers? (9, 10) The equation is 1 = (-1)

The equation is 1 = (-1) · 9 + 1 · 10 .

Which equations arise when the steps of the Euclidean algorithm are reversed to express the greatest common divisor of each of these pairs of integers as a linear combination of these integers? (36, 48) The equation is

The equation is 12 = (-1) · 36 + 48

Identify the correct statement about the given integers. 19, 41, 49, 64

The integers are pairwise relatively prime because no two of them have a common prime factor.

Find the sum and product of each of these pairs of numbers. Express your answers as octal expansions. The sum of the numbers (1111)8 and (776)8

The sum of the numbers (1111)8 and (776)8 is ( 2107 )8 and their product is ( 1106556 )8.

Find the sum and product of each of these pairs of numbers. Express your answers as binary expansions. The sum of the numbers (10 0000 0001)2 and (100 0000 0000)2 is

The sum of the numbers (10 0000 0001)2 and (100 0000 0000)2 is 1. ( 11000000001 )2 and their product is 2. ( 10000000010000000000 )

Identify the correct statement about the given integers. 21, 34, 55

These are pairwise relatively prime because no two of these integers share a prime factor.

Identify the correct statement about the given integers. 14, 25, 85

These integers are not pairwise relatively prime since two of them have a common prime factor 5.

Identify the correct statement about the given integers. 17, 28, 19, 23

These integers are pairwise relatively prime because no two have a common prime factor. The integers 17, 28, 19, and 23 are pairwise relatively prime because 17, 19, and 23 are prime and 28 = 7 · 22. Hence, no pair of these numbers have a common prime factor

Identify the correct steps involved in showing that if 2^n − 1 is prime, then n is prime. (Check all that apply.)

We will prove by contrapositive. Suppose n is not prime. Then, n= ab, for some integers a > 1 and b > 1. We must prove that 2ab − 1 is not prime. Consider the identity 2ab − 1 = (2a − 1) ⋅ (2a(b − 1) + 2a(b − 2) + ... + 2a + 1). The identity is valid, since we can clearly see on the right-hand side that all terms except 2ab and -1 cancel. Clearly, (2a(b − 1) + 2a(b − 2) + ... + 2a + 1) is greater than 1. Sincea > 1, the factor 2a − 1 is greater than 1. Since 2n − 1 is the product of two integers that are greater than 1, 2n − 1 is not prime.

Identify the correct step to prove that if a is an integer other than 0, then a divides 0.

a | 0 since 0 = a • 0