Answer :

The linear combination of gcd for 139 and (-191) is given by 16 * 139 + 11 * (-191) = 1. This means that the linear combination is 16 times 139 subtracted from 11 times -191. In other words, 16 times 139 is added to 11 times 191 to give us a result of 1. This linear combination helps us in finding the gcd of 139 and -191.

Given the numbers 139 and -191, we need to find their gcd and express it as a linear combination of the two numbers. First, we find the gcd using the Euclidean algorithm. We divide 191 by 139, getting a quotient of 1 and a remainder of 52. Then we divide 139 by 52, getting a quotient of 2 and a remainder of 35. Then we divide 52 by 35, getting a quotient of 1 and a remainder of 17. Then we divide 35 by 17, getting a quotient of 2 and a remainder of 1. Since the remainder is 1, we stop. The gcd of 139 and -191 is 1. Now we need to express 1 as a linear combination of 139 and -191. We use the extended Euclidean algorithm to find the coefficients of the linear combination. We start with the last two remainders and work backwards. Since the last remainder is 1, we have:

1 = 35 - 2 * 17.

Substituting 17 as 52 - 1 * 35, we get:

1 = 35 - 2 * (52 - 1 * 35) = 3 * 35 - 2 * 52.

Substituting 35 as 139 - 2 * 52, we get:

1 = 3 * (139 - 2 * 52) - 2 * 52 = 3 * 139 - 8 * 52.

Substituting 52 as -191 - 3 * 139, we get:

1 = 3 * 139 - 8 * (-191 - 3 * 139) = 25 * 139 + 8 * 191.

Therefore, the linear combination of gcd for 139 and -191 is 25 * 139 + 8 * (-191) = 1.

In conclusion, the linear combination of gcd for 139 and -191 is 25 * 139 + 8 * (-191) = 1. This means that 1 can be expressed as a linear combination of 139 and -191, where 139 is multiplied by 25 and -191 is multiplied by 8. The extended Euclidean algorithm is used to find the coefficients of the linear combination, which involves working backwards from the last two remainders obtained from the Euclidean algorithm.

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