Difference between revisions of "2006 AMC 10A Problems/Problem 22"
(→Solution) |
m (wikified and minor edit) |
||
| Line 3: | Line 3: | ||
<math>\mathrm{(A) \ } $5\qquad\mathrm{(B) \ } $10\qquad\mathrm{(C) \ } $30\qquad\mathrm{(D) \ } $90\qquad\mathrm{(E) \ } $210\qquad</math> | <math>\mathrm{(A) \ } $5\qquad\mathrm{(B) \ } $10\qquad\mathrm{(C) \ } $30\qquad\mathrm{(D) \ } $90\qquad\mathrm{(E) \ } $210\qquad</math> | ||
| + | |||
| + | |||
== Solution == | == Solution == | ||
| − | The problem can be restated as an equation of the form 300''p'' + 210''g'' = ''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 | + | The problem can be restated as an equation of the form 300''p'' + 210''g'' = ''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 [[integer]]s, 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 divisor]] of 300 and 210, which is, of course, 30 (c) |
| + | |||
== See also == | == See also == | ||
*[[2006 AMC 10A Problems]] | *[[2006 AMC 10A Problems]] | ||
Revision as of 14:32, 18 July 2006
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 divisor of 300 and 210, which is, of course, 30 (c)
