Question

Find the simplest possible form of each of the following numbers, module 5
1. -32
2. -35
3. -63

Answer 1

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. \(-63 \equiv 2 \mod 5\)