Answer to Question #80976 in Combinatorics | Number Theory for kenisha

Question

1. Theorem Let a,b and c be integers with a and b not both 0. If x = x0, y = y0 is an integer solution to the equation ax + by = c (that is, ax0 + by0 = c, then for every integer k, the numbers x = x0 + kb0 (a,b) and y = y0 − ka (a,b)
are integers that also satisfy the linear Diophantine equation ax + by = c. Moreover, every solution to the linear Diophantine equation ax + by = c is of this form.

2. Exercise Find all integer solutions to the equation 24x + 9y = 33.

3. Theorem Let a and b be integers with a,b > 0. Then gcd(a,b)· lcm(a,b) = ab.