2006 AMC 10A Problems/Problem 22
Problem
Two farmers agree that pigs are worth $300 and that goats are worth $210. When one farmer owes the other money, he pays the debt in pigs or goats, with "change" received in the form of goats or pigs as necessary. (For example, a $390 debt could be paid with two pigs, with one goat received in change.) What is the amount of the smallest positive debt that can be resolved in this way?
$\mathrm{(A) \ } $5\qquad\mathrm{(B) \ } $10\qquad\mathrm{(C) \ } $30\qquad\mathrm{(D) \ } $90\qquad\mathrm{(E) \ } $210\qquad$ (Error compiling LaTeX. Unknown error_msg)
Solution
The problem can be restated as an equation of the form 300p + 210g = x, where p is the number of pigs, g is the number of goats, and x is the positive debt. The problem asks us to find the lowest x possible. p and g must be integers, which makes the equation a Diophantine equation. Anyone with knowledge of number theory would know that there are integer solutions to a Diophantine equation if it is in the form ax + by = c, where c is the greatest common denominator of a and b. Therefore, the answer is simply the greatest common denominator of 300 and 210, which is, of course, 30 (c)
