3. Suppose that a and b are integers, a ≡ 4 (mod 13), and b ≡ 9 (mod 13). Find the integer cwith 0 ≤ c ≤ 12 such thata) c ≡ 9a (mod 13). b) c ≡ 11b (mod 13). c) c ≡ a + b (mod 13).d) c ≡ 2a + 3b (mod 13). e) c ≡ a2 + b2 (mod 13). f ) c ≡ a3 − b3 (mod 13).4. Evaluate these quantities.a) 13 mod 3 b) −97 mod 11 c) 155 mod 19 d) −221 mod 23

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

Let t∈Z. Which of the following describes values that are congruent to 7 (mod 12)?a.t+7b.7t+12c.36t-5d.12t+5

d) if a : b = 3 : 6 and b : c = 15 : 7,then a : b : c =

1. Does 17 divide each of these numbers?a) 68 b) 84 c) 357 d) 10012. What are the quotient and remainder whena) 19 is divided by 7? b) −111 is divided by 11? c) 789 is divided by 23?d) 1001 is divided by 13? e) 0 is divided by 19? f ) 3 is divided by 5?

A three digit number 4a3 is added to another three digit number 984 to give a four digit number 13b7, which is divisible by 11, then find the value of a.