Question

solve the Diophantine equations: 30x +47y = -11 and 102x + 1001y = 1

Answer 1

\[x=1843/4206\]
\[y=-91890/2143\]

Answer 2

For the equation 102x + 1001y = 1, you know that the gcd of 102 and 1001 is 1.
Therefore, there is some linear combination of 102 and 1001 that adds to 1.
Use the Euclidean Algorithm to find the coefficients m, n so that
102 n + 1001 m = 1.
Once you find that m and n, note that you can generate an infinite set of solutions by noting:
102 (n + 1001) + 1001 (m - 102) = 1 is always a true statement
if n and m is a solution to 102 n + 1001 m = 1.

Also, 102 (n + 2*1001) + 1001 (m - 2*102) = 1 is true.
102 (n + 1001k) + 1001 (m - 102)k = 1 for any integer k is true as well.

So we have an infinite set of solutions based on our initial solution given by the Euclidean algorithm to find the gcd of two numbers.

For 30x + 47 y = -11, the same idea holds.
We know that gcd)30, 47) = 1.
If we can find two numbers m, n such that 30m + 47n = 1,
then we can multiply both of those by -11 to get 30 M + 47N = -11.

And then we can form an infinite set of solutions by noting that
30 (M + 47k) + 47(N - 30k) = -11 would always be a true statement.

Answer 3

For the equation 102x + 1001y = 1, you know that the gcd of 102 and 1001 is 1.
Therefore, there is some linear combination of 102 and 1001 that adds to 1.
Use the Euclidean Algorithm to find the coefficients m, n so that
102 n + 1001 m = 1.
Once you find that m and n, note that you can generate an infinite set of solutions by noting:
102 (n + 1001) + 1001 (m - 102) = 1 is always a true statement
if n and m is a solution to 102 n + 1001 m = 1.

Also, 102 (n + 2*1001) + 1001 (m - 2*102) = 1 is true.
102 (n + 1001k) + 1001 (m - 102)k = 1 for any integer k is true as well.

So we have an infinite set of solutions based on our initial solution given by the Euclidean algorithm to find the gcd of two numbers.

For 30x + 47 y = -11, the same idea holds.
We know that gcd)30, 47) = 1.
If we can find two numbers m, n such that 30m + 47n = 1,
then we can multiply both of those by -11 to get 30 M + 47N = -11.

And then we can form an infinite set of solutions by noting that
30 (M + 47k) + 47(N - 30k) = -11 would always be a true statement.