Answer
15
Work Step by Step
We have to show that 15 is an inverse of 7 modulo 26.
Here, a=7 and b=26
By using Eucliden Algorithm:
26=7*3+5
7=5*1+2
5=2*2+1
2=1*2+0
gcd(a,b)=gcd(7,26)=1
Now:
1=5-2*2
1=5-2(7-5*1)
1=5*1-2*7+2*5
1=5(1+2)-2*7
1=5(3)-2*7
we can also write it as:
1=3(5)-2*7
1=3(26-7*3)-2*7
1=3*26-7*3*3-2*7
1=3*26-7(3*3*2)
1=3*26-7(11)
General form of this equation is: d=ax+by
So:
1=3(26)+7(-11)
or
1=26(3)+7(-11)
Now:
=inverse of 7 modulo 26
=-11(mod 26)
= 26-11 (mod 26)
=15 (mod 26)
=15
So, 15 in the inverse of our question.
It's proved.
