HW WK 5: Number theory

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

-111 is divided by 11. (quotient, remainder) =

(-11,10)

19 is divided by 7. (quotient, remainder) =

(2,5)

0 is divided by 19. (quotient, remainder) =

0, 0

2 is divided by 5. (quotient, remainder) =

0, 2

4 is divided by 19. (quotient, remainder) =

0,4

-15 mod 2 =

1

148 mod 7 =

1

The binary expansion of (135AD)16 is

1 0011 0101 1010 1101

321 The binary expansion of 321 is

1 0100 0001

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? (9999, 11111) (Check all that apply.)

1 = 2 · 5 − 9 1 = 247 · 1103 − 245 · 1112 1 = 2468 · 9999 − 2221 · 11111

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

6 is divided by 5. (quotient, remainder) =

1,1

The binary expansion of (80D)16 is

1000 0000 1101

The greatest prime factor of 107 is

107

1023 The binary expansion of 1023 is

11 1111 1111

The binary expansion of (CBBA)16 is

1100 1011 1011 1010

The binary expansion of (DAFACED)16 is

1101 1010 1111 1010 1100 1110 1101

The binary expansion of (DCFACED)16 is

1101 1100 1111 1010 1100 1110 1101

The greatest prime factor of 169 is

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? (28, 42) The equation is

14 = (-1) * 28 + 42

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? (45, 60) The equation is

15= (-1) * 45 +60

Identify the positive integers that are not relatively prime to 12. (Check all that apply.)

2 3 10

The decimal expansion of (1 1001)2 is

25

Find the least common multiple of each of these pair of integers. 2^1 · 7^1 and 5^1 · 13^1

2^1 · 5^1 · 7^1 · 13^1

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

2^4 · 3^3 · 5^4

Find the least common multiple of each of these pair of integers. 2^5 · 3^8 · 5 · 7 · 11^1 · 13 and 2^6 · 3^2 · 11^3 · 17^14

2^6 · 3^8 · 5 · 7 · 11^3 · 13 · 17^14

The smallest prime factor of 1147 is

31

The decimal expansion of (111 1100 0001 1101)2 is

31773

194 mod 19 =

4

List five integers that are congruent to 4 modulo 10:

4, 24, 44, 64, and 84

The decimal expansion of (01 1011 0101)2 is

437

The decimal expansion of (11 0011 1110)2 is

830

-95 mod 13 =

9

The decimal expansion of (11 1011 1110)2 is

958

Find the least common multiple of each of these pair of integers. 0 and 9

Does not exist

Identify the correct statement about the given integers. 27, 41, 49, 64

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

The statement "if a | bc, where a, b, and c are positive integers and a ≠ 0, then a | b or a | c" is (BLANK 1) Identify the correct statement to justify your answer. (Blank 2)

The statement "if a | bc, where a, b, and c are positive integers and a ≠ 0, then a | b or a | c" is blank 1: False . Identify the correct statement to justify your answer. Blank 2: Using the counterexample 4 |(2 × 2) but 4 does not divide 2, the given statement is false.

Click and drag the given steps (on the right) to their corresponding step names (on the left) to prove that if a | b and b | c, then a | c.

step 1: Suppose a | b, so that b = at for some t , and b | c, so that c = bs for some s. By definition of divisibility, step 2: We substitute the equation b=at into c=bs and get c=ats step 3: By definition of divisibility, c= a(ts), with ts being an integer, implies