2006 AMC 10A Problem 22

Problem 22 of 25HarderNumber Theory

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?

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Solution

A resolvable debt is

D = 300p + 210g

for integers

p,

g,

where a negative value means change received. Since

D = 30(10p + 7g)

and

\gcd(10, 7) = 1,

the value

10p + 7g

can be any integer, so

D

is any multiple of

30.

The smallest positive one is

30,

achieved by

30 = 300(-2) + 210(3)

(give

3

goats, receive

2

pigs). Thus, the correct answer is C.

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