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Pig and goat currency

Source: AMC 12 2006A, Problem 14

February 5, 2006
rationumber theorygreatest common divisorAMC

Problem Statement

Two farmers agree that pigs are worth $300 \$300 and that goats are worth $210 \$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 \$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? <spanclass=latexbold>(A)</span>$5<spanclass=latexbold>(B)</span>$10<spanclass=latexbold>(C)</span>$30<spanclass=latexbold>(D)</span>$90<spanclass=latexbold>(E)</span>$210 <span class='latex-bold'>(A) </span> \$5\qquad <span class='latex-bold'>(B) </span> \$10\qquad <span class='latex-bold'>(C) </span> \$30\qquad <span class='latex-bold'>(D) </span> \$90\qquad <span class='latex-bold'>(E) </span> \$210