Use the Euclidean algorithm to find the multiplicative inver

Use the Euclidean algorithm to find the multiplicative inverse of 8 mod 33. (Please) Use your answer to part 3a to help you solve 40x = 55(mod 165). (Please)

Solution

a)

33=8*4+1

1=33-8*4

So multiplicative inverse of 8 mod 33 is -4=33-4=29

So, 8*29=1 mod 33

(b)

165=33*5

40x=55 mod 165

40x-55 =0 mod 165

5(8x-11)=0 mod 165

So, 5(8x-11)=0 mod 33*5

So, 5*(8x-11)=0 mod 33

8x-11=0 mod 33

8x=11 mod 33

But, 8x=1 mod 33

So no solutions are possible