26. Multiply the month number of a student's birthday by 31 and the day number by 12, then add the two products together, the sum is 376. What is the student's birthday?
26. Multiply the month number of a student's birthday by 31 and the day number by 12, then add the two products to get a sum of 376. What is the student's birthday?
【Analysis】Assume the student's birthday is $X$ month $Y$ day. According to the problem, we have the equation $31 X + 12 Y = 376$. The positive integer solutions to this equation are the birthday, so the problem is to find the positive integer solutions to the indeterminate equation.
【Solution】Solution: Assume the student's birthday is $X$ month $Y$ day. According to the problem, we have the equation $31 X + 12 Y = 376$, which can be rewritten as $31X = 376 - 12Y$ (where $X$ and $Y$ are positive integers, $X$ is between 1 and 12, and $Y$ is between 1 and 31).
Since the right side of the equation can be divided by 4, the left side must also be divisible by 4. Since 4 and 31 are coprime, $X$ must be divisible by 4. Therefore, $X$ can only be $4, 8,$ or $12$. Upon inspection, only $X = 4$ satisfies the equation, at which point $Y = 21$. Therefore, the student's birthday is April 21. Answer: The student's birthday is April 21.