2006 AMC 10B Problem 16

P16-20~3 minPrint

Clocks, Calendars & Time Modular Arithmetic Set up the equation/formula and compute; no special trick needed

Leap Day, February $29$ , $2004$ , occurred on a Sunday. On what day of the week will Leap Day, February $29$ , $2020$ , occur?

Show solution without answering

Key insight

A normal year moves the weekday 1 forward; a year with a leap day moves it 2. That is 12 + 8 = 20, which is 6 past Sunday.

Solution

See how one year moves the weekday. A normal year has $365$ days, and $365 = 52 \cdot 7 + 1$ . So after $52$ full weeks there is $1$ day left over, and the same date lands one weekday later.
If the year you move through includes a February $29$ , it has $366$ days, which is $2$ days past a whole number of weeks. The weekday moves $2$ later.
Going from February $29$ , $2004$ to February $29$ , $2020$ is $16$ one-year steps. List the leap days you pass through: February $29$ of $2008$ , $2012$ , $2016$ and $2020$ . That is $4$ steps that move by $2$ , and the other $12$ steps move by $1$ .
Total shift: $12 \cdot 1 + 4 \cdot 2 = 20$ days forward.
Every $7$ days the weekday repeats, so take away whole weeks: $20 - 14 = 6$ . Six days after Sunday is Saturday.

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

Why this works

For weekday questions only the leftover after whole weeks matters, so think of a year as " $1$ extra day" (or $2$ for a leap year) instead of $365$ days. The careful part is counting leap days: count every February $29$ after the starting date, up to and including the ending date.

Alternative approach

Count the days directly: $16 \cdot 365 + 4 = 5844$ . Now $7 \cdot 835 = 5845$ , so $5844$ is one day short of a whole number of weeks. The weekday is one day before Sunday, which is Saturday.

Concise solution

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

Moving a date forward one ordinary year adds $365 = 52 \cdot 7 + 1$ days, which pushes the weekday forward by $1$ . If the year crossed contains a February $29$ , the jump is $366$ days and the weekday moves forward by $2$ .

From February 29, 2004 to February 29, 2020 is $16$ years. The leap days passed along the way are February 29 of $2008$ , $2012$ , $2016$ , and $2020$ : four of them (none of these years is a century exception). So $4$ of the $16$ yearly steps shift by $2$ and the other $12$ shift by $1$ :
$12 \cdot 1 + 4 \cdot 2 = 20 \equiv 6 \pmod 7.$

Six days after Sunday is Saturday (equivalently, one day back).

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

Review in the AoPS books

Introduction to Number Theory, Chapter 12: Introduction to Modular Arithmetic
Introduction to Number Theory, Chapter 10: Units Digits
Introduction to Number Theory, Chapter 14: Linear Congruences (§14.4)