Better question: "Given two integers $a$ and $b$, find the smallest number that can be expressed as a linear combination of the two."
I gave my students the better question with the numbers 28 and 68. They came up with two excellent approaches:
- List the multiples of both 28 and 68. Find the multiples that are closest together. (See photo)
- The multiples of 28 that are closest to 68 are $56 = 2\times 28 $ and $84 = 3\times 28$. We can find the difference of the differences of these multiples with 68: $68-56=12$, $84-68=16$, and one more difference gives $16-12=4$, our answer.