Practice question · Sort into groups
Congruence mod 4 splits the integers into four residue classes. Sort each number into its class.
Groups: Remainder 0 · Remainder 1 · Remainder 2 · Remainder 3
- 100
- 17
- 26
- -1
- 31
- 12
Hints
- Divide by 4 and keep only what is left over.
- -1 sits one step below 0, which puts it one step below the next multiple of 4, so its remainder is 3, not -1.
Show the answer
Remainder 0: 12, 100
Remainder 1: 17
Remainder 2: 26
Remainder 3: 31, -1
Why
12 and 100 are multiples of 4; 17 = 4(4) + 1; 26 = 4(6) + 2; 31 = 4(7) + 3; and -1 = 4(-1) + 3. There are exactly 4 classes, and every integer, negative ones included, lands in one of them. That partition is what makes congruence an equivalence relation.
Practise Modular Arithmetic
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