hint Use the Euclidean AlgorithmSolution1 3575119717 5735341

hint: Use the Euclidean Algorithm

Solution

1)

35=7*5,119=7*17

5*7=35=34+1=17*2+1

So, 5*7-17*2=1

So, 7(5*7-17*2)=7

35*7-119*2=7

gcd(7,79)=1 so there exist

Hence, 1 belongs to J. Hence, J=(1),b=1

2)

Using Euclidean Algo we can find:x,y so that:

7x+79y=1

We can use Euclidean algorithm but since 7 is a smaller number can do it manually which is easier for smaller numbers.

79=2 mod 7

Hence,79*3=3*2=6=-1 mod 7

79*3=237=238-1=7*34-1

Hence, 1=7*34-79*3

But, 35*7-119*2=7

So, 1=(35*7-119*2)*34-79*3

1=35*238-119*68-79*3