5. Find a div m and a mod m whena) a = −111, m = 99. b) a = −9999, m = 101.c) a = 10299, m = 999. d) a = 123456, m = 1001.6. Decide whether each of these integers is congruent to 5 modulo 17.a) 80 b) 103 c) −29 d) −1227. Find each of these values.a) (992 mod 32)3 mod 15 b) (34 mod 17)2 mod 11c) (193 mod 23)2 mod 31 d) (893 mod 79)4 mod 268. Convert the decimal expansion of each of these integers to a binary expansion.a) 23 b) 45 c) 241 d) 10259. Convert the binary expansion of each of these integers to a decimal expansion.a) (1 1011)2 b) (10 1011 0101)2c) (11 1011 1110)2 d) (111 1100 0001 1111)2
Question
- Find a div m and a mod m whena) a = −111, m = 99. b) a = −9999, m = 101.c) a = 10299, m = 999. d) a = 123456, m = 1001.6. Decide whether each of these integers is congruent to 5 modulo 17.a) 80 b) 103 c) −29 d) −1227. Find each of these values.a) (992 mod 32)3 mod 15 b) (34 mod 17)2 mod 11c) (193 mod 23)2 mod 31 d) (893 mod 79)4 mod 268. Convert the decimal expansion of each of these integers to a binary expansion.a) 23 b) 45 c) 241 d) 10259. Convert the binary expansion of each of these integers to a decimal expansion.a) (1 1011)2 b) (10 1011 0101)2c) (11 1011 1110)2 d) (111 1100 0001 1111)2
Solution
a) a = -111, m = 99. Dividing -111 by 99 gives -1 remainder -12. So, a div m = -1 and a mod m = 87 (because -12 mod 99 = 87).
b) a = -9999, m = 101. Dividing -9999 by 101 gives -99 remainder -0. So, a div m = -99 and a mod m = 0.
c) a = 10299, m = 999. Dividing 10299 by 999 gives 10 remainder 299. So, a div m = 10 and a mod m = 299.
d) a = 123456, m = 1001. Dividing 123456 by 1001 gives 123 remainder 333. So, a div m = 123 and a mod m = 333.
a) 80 is congruent to 5 modulo 17 because 80 mod 17 = 12. b) 103 is congruent to 5 modulo 17 because 103 mod 17 = 12. c) -29 is congruent to 5 modulo 17 because -29 mod 17 = 5. d) -122 is not congruent to 5 modulo 17 because -122 mod 17 = 3.
a) (992 mod 32)³ mod 15 = 8³ mod 15 = 512 mod 15 = 2. b) (34 mod 17)² mod 11 = 0² mod 11 = 0. c) (193 mod 23)² mod 31 = 4² mod 31 = 16. d) (893 mod 79)⁴ mod 26 = 5⁴ mod 26 = 625 mod 26 = 17.
a) 23 in binary is 10111. b) 45 in binary is 101101. c) 241 in binary is 11110001. d) 1025 in binary is 10000000001.
a) (1 1011)₂ in decimal is 23. b) (10 1011 0101)₂ in decimal is 1397. c) (11 1011 1110)₂ in decimal is 1918. d) (111 1100 0001 1111)₂ in decimal is 30783.
Similar Questions
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