Difference between revisions of "2019 AMC 8 Problems/Problem 14"

(Solution 1)
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Which clearly indicates if you start form a <math>x \equiv 3 \pmod{7}</math> you will not get a <math>y \equiv 0 \pmod{7}</math>.
 
Which clearly indicates if you start form a <math>x \equiv 3 \pmod{7}</math> you will not get a <math>y \equiv 0 \pmod{7}</math>.
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Any other starting value may lead to a <math>y \equiv 0 \pmod{7}</math>.
  
 
Which means our answer is <math>\boxed{\textbf{(C)}\ Wednesday}</math>.
 
Which means our answer is <math>\boxed{\textbf{(C)}\ Wednesday}</math>.

Revision as of 14:21, 20 November 2019

Problem 14

Isabella has $6$ 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 $10$ days until she has used them all. She knows that Pete's is closed on Sundays, but as she circles the $6$ 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?

$\textbf{(A) }$Monday$\qquad\textbf{(B) }$Tuesday$\qquad\textbf{(C) }$Wednesday$\qquad\textbf{(D) }$Thursday$\qquad\textbf{(E) }$Friday

Solution 1

Let $Day 1$ to $Day 2$ denote a day where one coupon is redeemed and the day when the second coupon is redeemed.

If she starts on a $Monday$ she redeems her next coupon on $Thursday$.

$Thursday$ to $Sunday$.

Thus $\boxed{\textbf{(A)}\ Monday}$ is incorrect.


If she starts on a $Tuesday$ she redeems her next coupon on $Friday$.

$Friday$ to $Monday$.

$Monday$ to $Thursday$.

$Thursday$ to $Sunday$.

Thus $\boxed{\textbf{(B)}\ Tuesday}$ is incorrect.


If she starts on a $Wednesday$ she redeems her next coupon on $Saturday$.

$Saturday$ to $Tuesday$.

$Tuesday$ to $Friday$.

$Friday$ to $Monday$.

$Monday$ to $Thursday$.

And on $Thursday$ she redeems her last coupon.


No sunday occured thus $\boxed{\textbf{(C)}\ Wednesday}$ is correct.


Checking for the other options,


If she starts on a $Thursday$ she redeems her next coupon on $Sunday$.

Thus $\boxed{\textbf{(D)}\ Thursday}$ is incorrect.


If she starts on a $Friday$ she redeems her next coupon on $Monday$.

$Monday$ to $Thursday$.

$Thursday$ to $Sunday$.


Checking for the other options gave us negative results, thus the answer is $\boxed{\textbf{(C)}\ Wednesday}$.

~phoenixfire


Solution 2

Let

$Sunday \equiv 0 \pmod{7}$

$Monday \equiv 1 \pmod{7}$

$Tuesday \equiv 2 \pmod{7}$

$Wednesday \equiv 3 \pmod{7}$

$Thursday \equiv 4 \pmod{7}$

$Friday \equiv 5 \pmod{7}$

$Saturday \equiv 6 \pmod{7}$


$10 \equiv 3 \pmod{7}$

$20 \equiv 6 \pmod{7}$

$30 \equiv 2 \pmod{7}$

$40 \equiv 5 \pmod{7}$

$50 \equiv 1 \pmod{7}$

$60 \equiv 4 \pmod{7}$


Which clearly indicates if you start form a $x \equiv 3 \pmod{7}$ you will not get a $y \equiv 0 \pmod{7}$.

Any other starting value may lead to a $y \equiv 0 \pmod{7}$.

Which means our answer is $\boxed{\textbf{(C)}\ Wednesday}$.

~phoenixfire

See Also

2019 AMC 8 (ProblemsAnswer KeyResources)
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

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