1. What time does a 24 hour clock read

    1. 100 hours after it reads 2:00?

      (100 + 2) mod 24 = 6:00

    2. 50 hours before it reads 12:00?

      (12 − 50) mod 24 = −38 mod 24 = 10 mod 24 = 10:00

    3. 268 hours after it reads 15:00?

      (268 + 15) mod 24 = 19:00

  2. Suppose that a and b are integers where a ≡ 11 (mod 19) and b ≡ 3 (mod 19). What is the value of integer c where 0 ≤ c ≤ 18 and

    1. c ≡ 12a (mod 19)

      (12)(11) mod 19 = 132 mod 19 = 18

      You can compute 132 mod 19 on a calculator as follows.

      1. Divide 132 by 19, getting 6.947368...
      2. Subtract the integer part, 6, getting 0.947368...
      3. Multiply by 19, getting 18.

      If you don't get an integer, you made a mistake.

    2. c ≡ a + b (mod 19)

      (11 + 3) mod 19 = 14

    3. c ≡ a2 + b2 (mod 19)

      112 mod 19 = 7
      32 mod 19 = 9
      (7 + 9) mod 19 = 16

    4. c ≡ a12 (mod 19)

      122 mod 19 = 144 mod 19
        = 11
      124 mod 19 = (112 mod 19)
        = 7
      128 mod 19 = (72 mod 19)
        = 11
      1212 mod 19 = (128 mod 19)(124 mod 19) mod 19
        = (11)(7) mod 19
        = 1

  3. Suppose that a and d are integers where d > 1. Show that the quotient q and remainder r when a is divided by d are q = ⌊a/d⌋ and r = a − d ⌊a/d⌋.

    1. It is clear that q and r are integers.

    2. The first requirement of the division theorem is that dq + r = a. Notice that

      dq + r = d ⌊a/d⌋ + a − d ⌊a/d⌋
        = a

    3. The second requirement of the division theorem is that 0 ≤ r < d. First, let's show that r ≥ 0.

      d ⌊a/d⌋ ≤ d(a/d)
        = a

      so

      r = a − d ⌊a/d⌋
        ≥ 0

      Now let's show that r < d.

      d⌊a/d⌋ > a/d − 1

      so

      r = a − d ⌊a/d⌋
        < a − d(a/d − 1)
        = d
  4. What are

    1. 144 mod 7

      4

    2. −17 mod 4

      3

    3. −101 mod 17

      1

      101 ≡ 16 (mod 17)

      −101 ≡ −16 (mod 17)
      ≡ 17 − 16 (mod 17)
      ≡ 1 (mod 17)

  5. List 5 integers that are congruent to 5 mod 12.

    5, 17, 29, 41, 53
  6. Convert each of the following from base 2 to base 10

    1. 10111

      23

    2. 110110110

      438

    3. 1010110101

      693
  7. Convert each of the following from base 16 to base 2.

    1. 123

      0001,0010,0011

    2. BADDFACEDD

      1011,1010,1101,1101,1111,1010,1110,1110,1101,1101
  8. Convert each of the following from base 10 to base 2.

    1. 45

      101101

    2. 1234

      10011010010

    3. 777

      1100001001