Whenever Deven and Laura owe each other money, they "pay" each other using stickers. They've agreed that a Harry Potter sticker is worth 49 dollars and a Twilight sticker is worth 35 dollars. They can even use stickers as "change" if one person overpays the other. For example, if Deven owes Laura 189 dollars, he can give her 6 Harry Potter stickers (6 * 49 = 294 dollars), and she can return 3 Twilight stickers ($3 * 35 = 105 dollars). This trade is like a transfer of 294-105=189 dollars. What is the smallest positive debt, in dollars, that can be paid off using sticker trading?
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To find the smallest positive debt that can be paid off using sticker trading, we need to find the least common multiple (LCM) of the values of the Harry Potter and Twilight stickers.
The value of a Harry Potter sticker is 49 dollars, and the value of a Twilight sticker is 35 dollars. To find the LCM, we can start by listing the multiples of each value until we find the first common multiple.
Multiples of 49: 49, 98, 147, 196, 245, 294, 343, ...
Multiples of 35: 35, 70, 105, 140, 175, 210, 245, ...
The first common multiple we see is 245. This means that the smallest positive debt that can be paid off using sticker trading is 245 dollars.
To verify this, let's see if we can find a combination of stickers to pay off this debt. If Deven owes Laura 245 dollars, he can give her 5 Harry Potter stickers (5 * 49 = 245 dollars), and she can give him 7 Twilight stickers (7 * 35 = 245 dollars). This trade is like a transfer of 245 - 245 = 0 dollars, which means they have settled the debt.