The Diophantine Hunter

Equations like 3x + 5y = 120 with 'x and y are positive integers,' followed by 'which of the following could be...'

Solving algebraically and then trying to reason abstractly. Faster: just enumerate.

1. Isolate one variable: e.g., y = (120 − 3x) / 5.
2. Find which integer values of x make y an integer (use divisibility).
3. List all valid (x, y) pairs systematically — there are usually 3-10 of them.
4. Compute the quantity asked for each pair and check against answer choices.
5. Don't forget to apply additional constraints (positive, distinct, range).

Multi-select 'which of the following could be' questions almost always come from this pattern.