Difference between revisions of "2013 AMC 12A Problems/Problem 17"
(→Solution 1) |
(→Solution 3 (Work backwards)) |
||
(12 intermediate revisions by 4 users not shown) | |||
Line 9: | Line 9: | ||
The first pirate takes <math>\frac{1}{12}</math> of the <math>x</math> coins, leaving <math>\frac{11}{12} x</math>. | The first pirate takes <math>\frac{1}{12}</math> of the <math>x</math> coins, leaving <math>\frac{11}{12} x</math>. | ||
− | The second pirate takes <math>\frac{2}{12}</math> of the remaining coins, leaving <math>\frac{10}{12} | + | The second pirate takes <math>\frac{2}{12}</math> of the remaining coins, leaving <math>\frac{10}{12}\cdot |
+ | \frac{11}{12}*x</math>. | ||
Note that | Note that | ||
− | <math>12^{11} = (2^2 | + | <math>12^{11} = (2^2 \cdot 3)^{11} = 2^{22} \cdot 3^{11}</math> |
− | <math>11! = 11 | + | <math>11! = 11 \cdot 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2</math> |
− | All the | + | All the <math>2</math>s and <math>3</math>s cancel out of <math>11!</math>, leaving |
− | <math>11 | + | <math>11 \cdot 5 \cdot 7 \cdot 5 = 1925</math> |
in the numerator. | in the numerator. | ||
Line 28: | Line 29: | ||
===Solution 2=== | ===Solution 2=== | ||
− | The answer cannot be an even number. | + | The answer cannot be an even number. Here is why: |
− | Consider the | + | Consider the highest power of 2 that divides the starting number of coins, and consider how this value changes as each pirate takes their share. At each step, the size of the pile is multiplied by some <math>\frac{n}{12}</math>. This means the highest power of 2 that divides the number of coins is continually decreasing or staying the same (except once, briefly, when we multiply by <math>\frac{2}{3}</math> for pirate 4, but it immediately drops again in the next step). |
− | Therefore, if the 12th pirate's coin total were even, | + | Therefore, if the 12th pirate's coin total were even, then it can't be the smallest possible value, because we can safely cut the initial pot (and all the intermediate totals) in half. We could continue halving the result until the 12th pirate's total is finally odd. |
− | Only one of the choices given is odd, <math>1925</math>. | + | Only one of the choices given is odd, <math>\boxed{\textbf{(D) }1925}</math>. |
+ | |||
+ | Note that if there had been multiple odd answer choices, we could repeat the reasoning above to see that the value also must not be divisible by 3. | ||
+ | |||
+ | ==Solution 3 (Work backwards)== | ||
+ | Let <math>x</math> be the number of coins the <math>12</math>th pirate takes. Then the number of coins the <math>k<12</math>th pirate takes is <math>\frac{12}{1} \cdot \frac{12}{2} \cdots \frac{12}{12-k} x</math>. For all these to be an integer, we need the denominators to divide into the numerators. Looking at prime factors, obviously there are sufficient <math>2</math>s and <math>3</math>s, so we just need <math>5^2 \cdot 7 \cdot 11 = \boxed{\textbf{(D) }1925}</math> to divide into <math>x</math>. -Frestho | ||
+ | |||
+ | ==Video Solution by Richard Rusczyk== | ||
+ | https://artofproblemsolving.com/videos/amc/2013amc12a/356 | ||
+ | |||
+ | ~dolphin7 | ||
== See also == | == See also == | ||
{{AMC12 box|year=2013|ab=A|num-b=16|num-a=18}} | {{AMC12 box|year=2013|ab=A|num-b=16|num-a=18}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 01:36, 11 October 2020
Contents
Problem 17
A group of pirates agree to divide a treasure chest of gold coins among themselves as follows. The
pirate to take a share takes
of the coins that remain in the chest. The number of coins initially in the chest is the smallest number for which this arrangement will allow each pirate to receive a positive whole number of coins. How many coins does the
pirate receive?
Solution
Solution 1
The first pirate takes of the
coins, leaving
.
The second pirate takes of the remaining coins, leaving
.
Note that
All the s and
s cancel out of
, leaving
in the numerator.
We know there were just enough coins to cancel out the denominator in the fraction. So, at minimum, is the denominator, leaving
coins for the twelfth pirate.
Solution 2
The answer cannot be an even number. Here is why:
Consider the highest power of 2 that divides the starting number of coins, and consider how this value changes as each pirate takes their share. At each step, the size of the pile is multiplied by some . This means the highest power of 2 that divides the number of coins is continually decreasing or staying the same (except once, briefly, when we multiply by
for pirate 4, but it immediately drops again in the next step).
Therefore, if the 12th pirate's coin total were even, then it can't be the smallest possible value, because we can safely cut the initial pot (and all the intermediate totals) in half. We could continue halving the result until the 12th pirate's total is finally odd.
Only one of the choices given is odd, .
Note that if there had been multiple odd answer choices, we could repeat the reasoning above to see that the value also must not be divisible by 3.
Solution 3 (Work backwards)
Let be the number of coins the
th pirate takes. Then the number of coins the
th pirate takes is
. For all these to be an integer, we need the denominators to divide into the numerators. Looking at prime factors, obviously there are sufficient
s and
s, so we just need
to divide into
. -Frestho
Video Solution by Richard Rusczyk
https://artofproblemsolving.com/videos/amc/2013amc12a/356
~dolphin7
See also
2013 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 16 |
Followed by Problem 18 |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.