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

Question

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

Answer 1

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. \(-75 \equiv 2 \mod 7\)