Note: These are my approaches to these problems. There are many ways to tackle each.
Most mathematicians would allow n\mathrel{∣}0 and (0,n) = n. Indeed, n\mathrel{∣}0 for all n is stated on the first page of one of the most respected number theory textbooks^{1} . So, alas, the “correct” response is to find out what standardized test uses which definition. This is yet another reason why standardized tests in higher mathematics are useless.
Compute the following using both the prime factorization method and the Euclidean algorithm:
Prime factorizations:
Euclidean algorithm:
So (720,241) = \mathbf{1}.
So (336,64) = 16.
So (-15,75) = 15.
Compute the least common multiples:
Find two integer solutions to each of the following, or state why no solutions exist:
Starting from the bottom and substituting for the previous remainder,
We find that 31 ⋅ 7 + 27 ⋅ (-8) = 1, so 31x - 27y = 11 has an initial solution of \mathbf{{x}_{0} = 7 ⋅ 11 = 77} and \mathbf{{y}_{0} = -1 ⋅-8 ⋅ 11 = 88}.
The general solutions have the form
Another solution is given by \mathbf{x(1) = 77 - 27 ⋅ 1 = 50} and \mathbf{y(1) = 88 - 31 ⋅ 1 = 57}.