Problem 3. The inhabitants of the Island of Misfortune, like us, divide the day into several hours, an hour into several minutes, and a minute into several seconds. But they have 77 minutes in a day and 91 seconds in a minute. How many seconds are there in a day on the Island of Misfortune?

[5 points] (I. V. Raskina)
Answer: 1001.

Solution: If you divide 77 by the number of minutes in an hour, you get the number of hours in a day. If you divide 91 by the number of minutes in an hour, you get the number of seconds in a minute. Therefore, both 77 and 91 are divisible by the number of minutes in an hour. Since there are obviously more than one minute in an hour, the number of minutes in an hour is 7—no other number greater than one divides both 77 and 91. Then, the number of hours in a day is $77: 7=11$ and $11 \cdot 91=1001$ seconds.

Comment: The same solution can be expressed algebraically. Specifically, let there be $a>1$ hours in a day, $b>1$ minutes in an hour, and $c>1$ seconds in a minute. Then $77=a b, 91=b c$. On the other hand, $77=7 \cdot 11$ and $91=7 \cdot 13$, and the numbers $7,11,13$ are prime, i.e., they cannot be factored further. Therefore, $b=7$, from which $a=11, c=13$. The total number of seconds in a day is $a b c=1001$.