2010 AMC 12B Problems/Problem 3
- The following problem is from both the 2010 AMC 12B #3 and 2010 AMC 10B #8, so both problems redirect to this page.
Problem
A ticket to a school play cost dollars, where
is a whole number. A group of 9th graders buys tickets costing a total of $
, and a group of 10th graders buys tickets costing a total of $
. How many values for
are possible?
Solution
We find the greatest common factor of and
to be
. The number of factors of
is
which is the answer
.
Solution 2
Since the price of one ticket is a whole number, it must be a divisor of both and
. An integer is a common divisor of two numbers if and only if it is divisible by their difference. Hence, all the possible prices are positive divisors of
, or
,
,
,
, and
, making the answer
, or C.
Video Solution
~Education, the Study of Everything
Video Solution
https://youtu.be/I3yihAO87CE?t=179
~IceMatrix
See also
2010 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 2 |
Followed by Problem 4 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
2010 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.