Question

С конечной остановки одновременно выезжают по двум маршрутам автобусы 1 возвращается каждые 45 минут 2 каждые 60 минут через Какое наименьшее время они снова окажутся на конечной остановке вместе

Answer 1

Через 3 часа   4*45 = 180 первый    60*3=180 второй 180мин это 3 часа

Answer 2

Problem Analysis

We are given that buses from two different routes depart simultaneously from the final stop. One bus returns every 45 minutes, while the other returns every 60 minutes. We need to determine the minimum time it takes for both buses to be back at the final stop together.

Solution

To find the minimum time for both buses to be back at the final stop together, we need to find the least common multiple (LCM) of 45 and 60. The LCM is the smallest positive integer that is divisible by both 45 and 60.

To find the LCM of 45 and 60, we can use the prime factorization method or the division method. Let's use the division method to find the LCM.

Division Method for Finding LCM

1. Write down the two numbers, 45 and 60. 2. Divide each number by the smallest prime number that divides both numbers evenly. In this case, the smallest prime number is 2. - 45 ÷ 2 = 22 remainder 1 - 60 ÷ 2 = 30 remainder 0 3. Write down the quotient (the result of the division) next to each number. - 45 ÷ 2 = 22 remainder 1 (quotient = 22) - 60 ÷ 2 = 30 remainder 0 (quotient = 30) 4. Repeat steps 2 and 3 until both numbers become 1. - 22 ÷ 2 = 11 remainder 0 (quotient = 11) - 30 ÷ 2 = 15 remainder 0 (quotient = 15) - 11 ÷ 11 = 1 remainder 0 (quotient = 1) - 15 ÷ 3 = 5 remainder 0 (quotient = 5) - 5 ÷ 5 = 1 remainder 0 (quotient = 1) 5. Multiply all the prime numbers and the quotients obtained in step 3. - 2 × 2 × 3 × 5 × 11 = 660

Therefore, the LCM of 45 and 60 is 660. This means that the two buses will be back at the final stop together after 660 minutes.

Answer

The two buses will be back at the final stop together after 660 minutes.

Verification

Let's verify this result using the given information. We know that one bus returns every 45 minutes and the other bus returns every 60 minutes. We need to check if both buses will be back at the final stop together after 660 minutes.

To verify, we can divide 660 by 45 and 60 and check if the remainders are 0.

- 660 ÷ 45 = 14 remainder 0 - 660 ÷ 60 = 11 remainder 0

Since both remainders are 0, it confirms that both buses will be back at the final stop together after 660 minutes.

Conclusion

The two buses will be back at the final stop together after 660 minutes.