1997 CEMC Pascal Problems/Problem 8

Problem

The greatest number of Mondays that can occur in the first $45$ days of a year is

$\text{ (A) }\ 5 \qquad\text{ (B) }\ 6 \qquad\text{ (C) }\ 7 \qquad\text{ (D) }\ 8 \qquad\text{ (E) }\ 9$

Solution 1

We can note that the greatest number of Mondays will occur when the first day of the year is a Monday.

We can then list the possible days (since Monday occurs every 7 days):

$1, 8, 15, 22, 29, 36, 43$

We cannot go past $43$ , as that goes outside the first $45$ days of the theoretical year. Counting the number of days, we have $\boxed {\textbf {(C) } 7}$ .

~anabel.disher

Solution 1.5

We can also notice that this problem can be solved by using the ceiling function. Since Monday repeats every 7 days, the number of Mondays in the first $n$ days of the year will just be $\lceil \frac{n}{7} \rceil$ .

Plugging in $45$ , we see:

$\lceil \frac{45}{7} \rceil$

Since $42 = 6 \times 7$ and $49 = 7 \times 7$ , and $45$ is between both $42$ and $49$ , the result in the division results in a number between $6$ and $7$ (exclusive). Thus, the ceiling is just $\boxed {\textbf {(C) } 7}$ .

~anabel.disher

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1997 CEMC Pascal Problems/Problem 8

Problem

Solution 1

Solution 1.5