Difference between revisions of "2019 AMC 8 Problems/Problem 14"
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== Solution 3 == | == Solution 3 == | ||
| − | Like Solution 2, let the days of the week be numbers <math>\pmod 7</math>. <math>3</math> and <math>7</math> are coprime, so continuously adding <math>3</math> to a number <math>\pmod 7</math> will cycle through all numbers from <math>0</math> to <math>6</math>. If a string of 6 numbers in this cycle does not contain <math>0</math>, then if you minus 3 from the first number of this cycle, it will always be <math>0</math>. So, the answer is <math>\boxed{\textbf{(C)}\ Wednesday}</math>. | + | Like Solution 2, let the days of the week be numbers<math>\pmod 7</math>. <math>3</math> and <math>7</math> are coprime, so continuously adding <math>3</math> to a number<math>\pmod 7</math> will cycle through all numbers from <math>0</math> to <math>6</math>. If a string of 6 numbers in this cycle does not contain <math>0</math>, then if you minus 3 from the first number of this cycle, it will always be <math>0</math>. So, the answer is <math>\boxed{\textbf{(C)}\ Wednesday}</math>. ~~SmileKat32 |
==See Also== | ==See Also== | ||
Revision as of 21:10, 20 November 2019
Problem 14
Isabella has 
coupons that can be redeemed for free ice cream cones at Pete's Sweet Treats. In order to make the coupons last, she decides that she will redeem one every 
days until she has used them all. She knows that Pete's is closed on Sundays, but as she circles the dates on her calendar, she realizes that no circled date falls on a Sunday. On what day of the week does Isabella redeem her first coupon?
Monday
Tuesday
Wednesday
Thursday
Friday
Solution 1
Let 
to 
denote a day where one coupon is redeemed and the day when the second coupon is redeemed.
If she starts on a 
she redeems her next coupon on 
.
to $\text{Sunday$ (Error compiling LaTeX. Unknown error_msg)}.
Thus 
is incorrect.
If she starts on a 
she redeems her next coupon on 
.
to 
.
to 
.
to 
.
Thus 
is incorrect.
If she starts on a 
she redeems her next coupon on 
.
to .
to .
to .
to .
And on she redeems her last coupon.
No sunday occured thus 
is correct.
Checking for the other options,
If she starts on a she redeems her next coupon on .
Thus 
is incorrect.
If she starts on a she redeems her next coupon on .
to .
to .
Checking for the other options gave us negative results, thus the answer is .
~phoenixfire
Solution 2
Let
Which clearly indicates if you start form a 
you will not get a 
.
Any other starting value may lead to a .
Which means our answer is .
~phoenixfire
Solution 3
Like Solution 2, let the days of the week be numbers
. 
and 
are coprime, so continuously adding to a number will cycle through all numbers from 
to . If a string of 6 numbers in this cycle does not contain , then if you minus 3 from the first number of this cycle, it will always be . So, the answer is . ~~SmileKat32
See Also
| 2019 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 13 |
Followed by Problem 15 | |
| 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. 
