Multiplicative Inverse. Affine Cipher

[cks]

Here is how to find the a^(-1)

According to the definition,

aa^(-1)=1mod (26)

For example, let’s try a=3

According to Extended Euclidean Algorithm

gcd⁡(a,26)=gcd⁡(3,26)=gcd⁡(3,2)=gcd⁡(1,2)
Where
1=3-1*2
2=26-8*3
1=3-1*(26-8*3)=-1*26+9*3

With 9 found,
a^(-1)=9

However, to find a=5
gcd⁡(5,26)=gcd⁡(5,1)
1=1*26-5*5

So,a^(-1)=-5?

(-5)(5)=1mod(26) which is correct

How to get a^(-1) in this case as shown in the table which is 15?

 

[cks]

I manage to solve the problems. Never mind. Thank you.

 

[HallsofIvy]

5^-1 (mod 26) is NOT 15. 5*15= 75= 2*26+ 23. 5*16= 23 (mod 26), not 1.

5^-1 (mod 26)= 26+ (-5)= 21. 5*21= 105= 4*26+ 1. 5*21= 1 (mod 26).

-5= 21 (mod 26).