Definition 5.1.3.1.
Let \(m\in\NN\) and \(a\in\ZZ_m\) and \(b\in\ZZ_m\text{.}\) If \(a\oplus_m b=0\) we call \(b\) an additive inverse of \(a\) modulo \(m\text{.}\)
Subsection 5.1.3 Modular inverses
Example 5.1.3.2. Additive inverses modulo \(6\).
We list additive inverse modulo \(6\text{.}\)
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An additive inverse modulo \(6\) of 0 is 0, because \(0\oplus_6 0=(0+0)\bmod 6=0\bmod 6=0\text{.}\)
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An additive inverse modulo \(6\) of 1 is 5, because \(1\oplus_6 5=(1+5)\bmod 6=6\bmod 6=0\text{.}\)
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An additive inverse modulo \(6\) of 2 is 4, because \(2\oplus_6 4=(2+4)\bmod 6=6\bmod 6=0\text{.}\)
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An additive inverse modulo \(6\) of 3 is 3, because \(3\oplus_6 3=(3+3)\bmod 6=6\bmod 6=0\text{.}\)
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An additive inverse modulo \(6\) of 4 is 2, because \(4\oplus_6 2=(4+2)\bmod 6=6\bmod 6=0\text{.}\)
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An additive inverse modulo \(6\) of 5 is 1, because \(5\oplus_6 1=(5+1)\bmod 6=6\bmod 6=0\text{.}\)
Definition 5.1.3.3.
Let \(m\in\NN\) and \(a\in\ZZ_m\) and \(b\in\ZZ_m\text{.}\) If \(a\otimes_m b=1\) we call \(b\) a multiplicative inverse of \(a\) modulo \(m\text{.}\)
Example 5.1.3.4. Multiplicative inverses modulo \(6\).
We list multiplicative inverses modulo \(6\text{.}\)
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There is no multiplicative inverse modulo 6 of 0, because \(0\otimes_6 a=(0\cdot a)\bmod 6=0\) for all \(a\in\ZZ_6\text{.}\)
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A multiplicative inverse modulo 6 of 1 is 1, because \(1\otimes_6 1=(1\cdot 1)\bmod 6=1\bmod 6=1\text{.}\)
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There is no multiplicative inverse modulo 6 of 2, because \(2\otimes_6 a=(2\cdot a)\bmod 6\ne 1\) for all \(a\in\ZZ_6\text{.}\)
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There is no multiplicative inverse modulo 6 of 3, because \(3\otimes_6 a=(3\cdot a)\bmod 6\ne 1\) for all \(a\in\ZZ_6\text{.}\)
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There is no multiplicative inverse modulo 6 of 4, because \(4\otimes_6 a=(4\cdot a)\bmod 6\ne 1\) for all \(a\in\ZZ_6\text{.}\)
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A multiplicative inverse modulo 6 of 5 is 5, because \(5\otimes_6 5=(5\cdot 5)\bmod 6=25 \bmod 6=1\text{.}\)
Definition 5.1.3.5.
Let \(m\in\NN\) and \(a\in\ZZ_m\text{.}\) When there exists \(b\in\ZZ_m-\{0\}\) such that \(a\otimes_m b=0\) then we call \(a\) a zero divisor modulo \(m\text{.}\)
Example 5.1.3.6. Zero divisors modulo \(6\).
We list zero divisors modulo 6.
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2 is a zero divisor modulo 6, because \(2\otimes_6 3=(2\cdot3)\bmod 6=6\bmod 6=0\text{.}\)
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3 is a zero divisor modulo 6, because \(3\otimes_6 2=(3\cdot2)\bmod 6=6\bmod 6=0\text{.}\)
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4 is a zero divisor modulo 6, because \(4\otimes_6 3=(4\cdot3)\bmod 6=12\bmod 6=0\text{.}\)
