Question

To find: The inverse for 210 modulo 13.

Polynomial factorization
ANSWERED
asked 2021-02-16
To find: The inverse for 210 modulo 13.

Answers (1)

2021-02-18

Given information:
210 modulo 13
Calculation:
Step 1. \(210=13 \cdot 16+2\) and so \(2=210-13 \cdot 16.\)
Step 2. \(13=2 \cdot 6+1\) and so \(1=13-2 \cdot 6\)
Step 3. \(6=1 \cdot 6+0\)
Thus, \(GCD(210,13)=1\)
Substituting back through step \(2-1\)
\(1=13-2 \cdot 6\)
\(=13-(210-13\cdot16)\cdot6\) by step (1)
\(=13-210\cdot6+13\cdot96\)
\(=13\cdot 97-210\cdot6\)
\(1=97\cdot13-6\cdot210\)
\(1=(-6)\cdot210+97\cdot13\)
Thus, \(\displaystyle{210}\cdot{\left(-{6}\right)}={1}{\left({b} \mod{13}\right)}\) by the definition of \(b mod\), so -6 is an inverse for \(\displaystyle{210}{b} \mod {13}\).
Conclusion:
-6 is an inverse for 210 modulo 13.

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