Unlike ordinary addition, there is not an additive inverse to each integer in modular arithmetic
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In modular arithmetic, we work with numbers that are taken modulo some fixed positive integer, usually denoted by m. The numbers that we work with in modular arithmetic are called residues.
When we perform addition in modular arithmetic, we add the residues and then take the result modulo m. For example, if we are working modulo 5 and we want to add 3 and 4, we compute:
3 + 4 = 7
But then we take this result modulo 5, which means we ask what is the remainder when 7 is divided by 5?. The answer is 2, so:
3 + 4 ≡ 2 (mod 5)
In modular arithmetic, we can also define a notion of additive inverse. The additive inverse of a residue r is another residue -r that, when added to r, gives 0 modulo m. For example, the additive inverse of 3 modulo 5 is 2, because:
3 + 2 = 5 ≡ 0 (mod 5)
However, in modular arithmetic, not every residue has an additive inverse. For example, in the integers modulo 6, the residue 2 does not have an additive inverse, because there is no residue that can be added to 2 to give 0 modulo 6.
This happens when the residue and the modulus share factors other than 1. In the example above, 2 and 6 are both divisible by 2, so we cannot find an additive inverse for 2. In general, a residue r will have an additive inverse modulo m if and only if r and m are relatively prime, i.e., their greatest common divisor is 1.
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