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Find the simplest possible form of each of the following numbers, module 5

1. -32
2. -35
3. -63

1 answer

To find the simplest possible form of each number modulo 5, we will calculate each number's remainder when divided by 5.

  1. For \(-32\): \[ -32 \mod 5 = -32 + 35 = 3 \] (since \(-32 + 35 = 3\) is the smallest non-negative integer that is congruent to \(-32\) mod 5.)

  2. For \(-35\): \[ -35 \mod 5 = -35 + 35 = 0 \] (since \(-35\) is already a multiple of 5.)

  3. For \(-63\): \[ -63 \mod 5 = -63 + 65 = 2 \] (since \(-63 + 65 = 2\) is the smallest non-negative integer that is congruent to \(-63\) mod 5.)

So the simplest possible forms for each number modulo 5 are:

  1. \(-32 \equiv 3 \mod 5\)
  2. \(-35 \equiv 0 \mod 5\)
  3. \(-63 \equiv 2 \mod 5\)