Difference between revisions of "2010 AMC 10A Problems/Problem 10"

Revision as of 09:30, 9 April 2019

Marvin had a birthday on Tuesday, May 27 in the leap year $2008$ . In what year will his birthday next fall on a Saturday?

$\mathrm{(A)}\ 2011 \qquad \mathrm{(B)}\ 2012 \qquad \mathrm{(C)}\ 2013 \qquad \mathrm{(D)}\ 2015 \qquad \mathrm{(E)}\ 2017$

2010 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 9	Followed by Problem 11
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All AMC 10 Problems and Solutions

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 </math>
-==Solution==
-<math>\boxed{(E)}</math> <math> 2017 </math>
-There are <math>365</math> days in a non-leap year. There are <math>7</math> days in a week. Since <math>365 = 52 \cdot 7 + 1</math> (or <math>365</math> is congruent to <math>1 \mod{ 7}</math>), the same date (after February) moves "forward" one day in the subsequent year, if that year is not a leap year.
-For example:
-<math>5/27/08</math>	Tue
-<math>5/27/09</math>	Wed
-However, a leap year has <math>366</math> days, and <math>366 = 52 \cdot 7 + 2</math> . So the same date (after February) moves "forward" '''two''' days in the subsequent year, if that year is a leap year.
-For example:
-<math>5/27/11</math>	Fri
-<math>5/27/12</math>	Sun
-You can keep count forward to find that the first time this date falls on a Saturday is in <math> 2017</math>:
-<math>5/27/13</math>	Mon
-<math>5/27/14</math>	Tue
-<math>5/27/15</math>	Wed
-<math>5/27/16</math>	Fri
-<math>5/27/17</math>	Sat