Find the least positive inverse for 43 modulo 660.

To find:

The least positive inverse for 43 modulo 660.

Given information:

43 mod 660

Concept used:

Inverse of a ​modulo n: For all integers a and n, if gcd(a,n)=1, then there exists an integer s

such that as≡1(modn). The integer s is called the inverse of a ​modulo n.

Calculation:

First check if gcd(660,43)=1.

Use Euclidean algorithm to find gcd(660,43).

Divide 660 by 43 to get quotient=15 and remainder=15.

Then, 660=43⋅15+15 and hence gcd(660,43)=gcd(43,15)

Divide 43 by 15 to get quotient=2 and remainder=13

Then, 43=15⋅2+13 and hence gcd(43,15)=gcd(15,13)

Divide 15 by 13 to get quotient=1 and remainder=2

Then, 15=13⋅1+2 and hence gcd(15,13)=gcd(13,2)

Divide 13 by 2 to get quotient=6 and remainder=1

Then, 13=2⋅6+1 and hence gcd(13,2)=gcd(2,1)

Divide 2 by 1 to get quotient=2 and remainder=0

Then, 2=1⋅2+0 and hence gcd(2,1)=gcd(1,0)

Now, gcd(1,0)=1.

Hence,

gcd(660,43)=gcd(43,15)=gcd(15,13)=gcd(13,2)=gcd(2,1)=gcd(1,0)=1

Since gcd(660,43)=1, the inverse for 43 modulo 660 exists.

Find the inverse for 43 modulo 660.

To do so, find a linear combination of 43 and​ 660 that equals 1.

Now,

660=43⋅15+15⇒15=660−43⋅15