2010 AMC 10A Problem 10

P1-10~3 minPrint

Clocks, Calendars & Time Modular Arithmetic Set up the equation/formula and compute; no special trick needed Compute small cases, spot the pattern, generalize

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

Show solution without answering

Key insight

Each year the birthday moves 1 weekday later, or 2 days when a February 29 comes between. Count forward from Tuesday until you land on Saturday.

Solution

First figure out how the weekday moves from one birthday to the next.

A normal year has $365$ days. That is $52$ full weeks ( $364$ days) plus $1$ extra day. So next year's birthday falls $1$ weekday later.
If a February 29 comes between two birthdays, there are $366$ days, which is $52$ weeks plus $2$ days. Then the birthday moves $2$ weekdays later. This happens when the later year is a leap year: 2012 and 2016. (February 29, 2008 is before May 27, 2008, so it does not count.)
Now step year by year from Tuesday in 2008:

Year	Move	Weekday
2009	$+1$	Wednesday
2010	$+1$	Thursday
2011	$+1$	Friday
2012 (leap)	$+2$	Sunday
2013	$+1$	Monday
2014	$+1$	Tuesday
2015	$+1$	Wednesday
2016 (leap)	$+2$	Friday
2017	$+1$	Saturday

Notice that in 2012 the birthday jumps from Friday straight to Sunday and skips Saturday. The first Saturday is in 2017.

The answer is $\boxed{\textbf{(E)}\ 2017}$ .

Why this works

Weekdays repeat every $7$ days, so only the leftover days after full weeks matter: $1$ for a normal year, $2$ for a leap year. The danger is the leap-year jump, which can skip right over the day you are looking for. Writing a small table keeps track of it.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

From one May 27 to the next is $365$ days, which is $52$ weeks plus $1$ day, so the weekday advances by $1$ . If a February 29 lies in between (that is, the later year is a leap year), the gap is $366$ days and the weekday advances by $2$ .

Saturday is $4$ days after Tuesday, so we need the total shift to be $4$ , or $11$ , or $18$ , and so on. Tally year by year, starting from Tuesday in $2008$ :

$2009$ : $+1$ , Wednesday
$2010$ : $+1$ , Thursday
$2011$ : $+1$ , Friday
$2012$ (leap): $+2$ , Sunday
$2013$ : $+1$ , Monday
$2014$ : $+1$ , Tuesday
$2015$ : $+1$ , Wednesday
$2016$ (leap): $+2$ , Friday
$2017$ : $+1$ , Saturday

The birthday first falls on a Saturday in $2017$ .

The answer is $\boxed{\textbf{(E)}\ 2017}$ .

Review in the AoPS books

Introduction to Number Theory, Chapter 12: Introduction to Modular Arithmetic
Prealgebra, Chapter 15: Problem-Solving Strategies (§15.1)
Introduction to Number Theory, Chapter 10: Units Digits
Introduction to Number Theory, Chapter 14: Linear Congruences (§14.4)