Question

某个有 $31$ 天的月份中，星期一和星期三的天数相同。这个月的第一天可能是七天中的多少种？

A month with $31$ days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month?

$2$

$3$

$4$

$5$

$6$

$31$ 除以 $7$ 余 $3$ ，所以从该月第一天开始的连续三个星期几会出现五次，其余出现四次。

要使星期一和星期三次数相同，它们不能恰好只有一个在这三个多出的星期几中。

检查可知，该月只能从星期一、星期四或星期五开始。

所以正确答案是 B。

Note that $31$ days leaves a remainder of $3$ when divided by $7.$

This means that if the month starts on a Saturday, Sunday, Tuesday, or Wednesday, there will be an uneven number of Mondays and Wednesdays.

Then the month can only start on a Monday, Thursday, or Friday.

Thus, B is the correct answer.