Difference between revisions of "2017 AMC 8 Problems/Problem 24"
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==Solution 1 (Least Common Multiple)== | ==Solution 1 (Least Common Multiple)== | ||
− | Note that <math>\mathrm(3,4,5)=60,</math> so there is a cycle every <math>60</math> days. | + | Note that <math>\mathrm{lcm}(3,4,5)=60,</math> so there is a cycle every <math>60</math> days. |
As shown below, all days in a cycle that Mrs. Sanders receives a phone call from any of her grandchildren are colored in red, yellow, or green. | As shown below, all days in a cycle that Mrs. Sanders receives a phone call from any of her grandchildren are colored in red, yellow, or green. |
Revision as of 15:32, 15 July 2021
Contents
Problem
Mrs. Sanders has three grandchildren, who call her regularly. One calls her every three days, one calls her every four days, and one calls her every five days. All three called her on December 31, 2016. On how many days during the next year did she not receive a phone call from any of her grandchildren?
Solution 1 (Least Common Multiple)
Note that so there is a cycle every
days.
As shown below, all days in a cycle that Mrs. Sanders receives a phone call from any of her grandchildren are colored in red, yellow, or green.
The year 2017 has
days, or
cycles and
days.
- For each cycle, there are
days that Mrs. Sanders does not receive a phone call, as indicated by the white squares.
- For the last
days, there are
days that Mrs. Sanders does not receive a phone call, as indicated by the first
days in a cycle.
Together, the answer is
~MRENTHUSIASM
Solution 2 (Detailed Version of Principle of Inclusion-Exclusion)
We use Principle of Inclusion-Exclusion. There are days in the year, and we subtract the days that she gets at least
phone call, which is
To this result we add the number of days where she gets at least phone calls in a day because we double subtracted these days. This number is
We now subtract the number of days where she gets three phone calls, which is . Therefore, our answer is
Solution 3 (Condensed Version of Principle of Inclusion-Exclusion)
For this problem, we use the Principle of Inclusion-Exclusion (PIE). Notice every day a day would be a multiple of
, every
day a day is a multiple of
, and every
days it's a multiple of
days without calls. Note that in the last five days of the year, days
and
also do not have any calls, as they are not multiples of
,
, or
. Thus our answer is
.
Video Solution
https://youtu.be/a3rGDEmrxC0 - Happytwin
https://youtu.be/Zhsb5lv6jCI?t=2797
See Also
2017 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
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 AJHSME/AMC 8 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.