Difference between revisions of "2006 AMC 12A Problems/Problem 14"
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==Solution 1== | ==Solution 1== | ||
− | The problem can be restated as an equation of the form <math>300p + 210g = x</math>, where <math>p</math> is the number of pigs, <math>g</math> is the number of goats, and <math>x</math> is the positive debt. The problem asks us to find the lowest ''x'' possible. ''<math>p</math>'' and ''<math>g</math>'' must be integers, which makes the equation a [[Diophantine equation]]. Bezout’s | + | The problem can be restated as an equation of the form <math>300p + 210g = x</math>, where <math>p</math> is the number of pigs, <math>g</math> is the number of goats, and <math>x</math> is the positive debt. The problem asks us to find the lowest ''x'' possible. ''<math>p</math>'' and ''<math>g</math>'' must be integers, which makes the equation a [[Diophantine equation]]. [[Bezout’s Lemma]] tells us that the smallest <math>c</math> for the Diophantine equation <math>am + bn = c</math> to have solutions is when <math>c</math> is the GCD ([[greatest common divisor]]) of <math>a</math> and <math>b</math>. Therefore, the answer is <math>gcd(300,210)=\boxed{\textbf{(C) }30}.</math> |
==Solution 2== | ==Solution 2== |
Revision as of 10:07, 19 December 2021
- The following problem is from both the 2006 AMC 12A #14 and 2006 AMC 10A #22, so both problems redirect to this page.
Problem
Two farmers agree that pigs are worth dollars and that goats are worth dollars. 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 dollar 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?
Solution 1
The problem can be restated as an equation of the form , where is the number of pigs, is the number of goats, and is the positive debt. The problem asks us to find the lowest x possible. and must be integers, which makes the equation a Diophantine equation. Bezout’s Lemma tells us that the smallest for the Diophantine equation to have solutions is when is the GCD (greatest common divisor) of and . Therefore, the answer is
Solution 2
Alternatively, note that is divisible by no matter what and are, so our answer must be divisible by . Since we want the smallest integer, we can suppose that the answer is and go on from there. Note that three goats minus two pigs gives us exactly. Since our supposition can be achieved, the answer is .
Solution 3
Let us simplify this problem. Dividing by , we get a pig to be: , and a goat to be . It becomes evident that if you exchange pigs for goats, we get the smallest positive difference - , since we can't make a non-integer with a linear combination of integers. Since we originally divided by , we need to multiply again, thus getting the answer:
See also
Bezout's Identity:
https://artofproblemsolving.com/wiki/index.php/Bezout%27s_Lemma
https://brilliant.org/wiki/bezouts-identity/
2006 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 13 |
Followed by Problem 15 |
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Preceded by Problem 21 |
Followed by Problem 23 | |
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