Question

# To find: The inverse for 210 modulo 13.

Polynomial factorization
To find: The inverse for 210 modulo 13.

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.