Find the simplest possible form of each of the following numbers for the given moduli

1. -17(mod 2)
2. -56(mod 13)
3. -50(mod4)
4. -75(mod7)

1 answer

To find the simplest possible form of each number under the given moduli, we need to calculate the equivalent positive remainder for each negative number.

  1. For -17 (mod 2):

    • First, we find the positive representative by adding the modulus until we get a non-negative number.
    • -17 + 2 = -15
    • -15 + 2 = -13
    • -13 + 2 = -11
    • -11 + 2 = -9
    • -9 + 2 = -7
    • -7 + 2 = -5
    • -5 + 2 = -3
    • -3 + 2 = -1
    • -1 + 2 = 1
    • Therefore, \(-17 \equiv 1 \mod 2\).
  2. For -56 (mod 13):

    • We will add 13 until we get a non-negative number.
    • -56 + 13 = -43
    • -43 + 13 = -30
    • -30 + 13 = -17
    • -17 + 13 = -4
    • -4 + 13 = 9
    • Therefore, \(-56 \equiv 9 \mod 13\).
  3. For -50 (mod 4):

    • Again, adding the modulus 4 until we get a non-negative number:
    • -50 + 4 = -46
    • -46 + 4 = -42
    • -42 + 4 = -38
    • -38 + 4 = -34
    • -34 + 4 = -30
    • -30 + 4 = -26
    • -26 + 4 = -22
    • -22 + 4 = -18
    • -18 + 4 = -14
    • -14 + 4 = -10
    • -10 + 4 = -6
    • -6 + 4 = -2
    • -2 + 4 = 2
    • Therefore, \(-50 \equiv 2 \mod 4\).
  4. For -75 (mod 7):

    • Adding 7 until we reach a non-negative number:
    • -75 + 7 = -68
    • -68 + 7 = -61
    • -61 + 7 = -54
    • -54 + 7 = -47
    • -47 + 7 = -40
    • -40 + 7 = -33
    • -33 + 7 = -26
    • -26 + 7 = -19
    • -19 + 7 = -12
    • -12 + 7 = -5
    • -5 + 7 = 2
    • Therefore, \(-75 \equiv 2 \mod 7\).

In summary, the simplified forms are:

  1. \(-17 \equiv 1 \mod 2\)
  2. \(-56 \equiv 9 \mod 13\)
  3. \(-50 \equiv 2 \mod 4\)
  4. \(-75 \equiv 2 \mod 7\)