mention multiplicative inverse of 5/13
Answers
Answer:
Step-by-step explanation:
As the field of remainders mod 13 is finite, we can search for the inverse by trial and error. There are only 11 numbers to try (excluding 0 and 1). But I’m lazy, and therefore I will use Fermat’s little theorem.
We know that
512≡1(mod13),
therefore, the inverse of 5 is 511 .
Let’s try to find it as a remainder mod 13:
511=510⋅5=(52)5⋅5
=255⋅5≡(−1)5⋅5
=−5≡8(mod13).
Therefore,
5−1=8(mod13).
Comprobation:
5⋅8=40=39+1≡1(mod13).
If the divisor is not prime, but coprime with the number given, instead of FLT use Euler’s theorem.
An alternative approach:
Providing the divisor and the number are coprime (if not, there is no solution), use Euclid algorithm:
13=3⋅5−2
5=2⋅2+1
⇒
1=5−2⋅2=5−2(3⋅5−13)
=−5⋅5+2⋅13≡−5⋅5(mod13).
Therefore, the inverse of 5 is −5≡8(mod13) .
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