Question

Мама хочет наказать Петю за двойку по математике. Они договорились о следующем.Петя задумывает двузначное число и сообщает его маме. После этого мама тожезадумывает двузначное число и называет его Пете. Дальше Петя в первую минутуприбавляет мамино число к своему числу, во вторую прибавляет мамино число кполученной сумме , в третью – к вновь полученной сумме и т.д. Если в течение двух часову него получится сумма, оканчивающаяся на две одинаковые цифры, мама отпустит егогулять. Сможет ли мама не позволить Пете в этот день погулять?

Answer 1

За 2 часа мамино число увеличится в 120 раз (так как одно сложение происходит за одну минуту. А 2 часа = 120 минут) 
Таким образом полученое число будет заканчиваться на 0 если число мамы не кратно 10 (например 24, 57, 17 и т.д.) и 00 если кратно 10 (например 20, 30, 80).
Таким образом единственный вариант для Пети пойти погулять - это загадать число кратное 11 (например 22, 55, 99), при условии, что мама загадает число кратное 10. 
Выглядеть это будет так: 
число Пети + 120* число мамы 
Пусть Петя загадал 44, а мама 80, тогда по истечению 2-х часов получится число 44 + 120*80 = 9644

Answer 2

Problem Analysis

In this problem, Petya and his mother have made an agreement. Petya thinks of a two-digit number and tells it to his mother. Then, Petya adds his mother's number to his own number repeatedly for two hours. If the final sum ends with two identical digits, Petya's mother will allow him to go out. The question is whether Petya's mother can prevent him from going out on that day.

Solution

To determine if Petya's mother can prevent him from going out, we need to analyze the possible outcomes of the addition process. Let's consider all the possible two-digit numbers Petya can think of and analyze the results.

Possible Two-Digit Numbers

Petya can think of any two-digit number between 10 and 99, inclusive. There are 90 possible two-digit numbers.

Addition Process

Petya adds his mother's number to his own number repeatedly for two hours. Let's analyze the possible outcomes of this process.

1. After the first addition, Petya's number will change. 2. After the second addition, Petya's number will change again. 3. This process continues for two hours, with Petya's number changing after each addition.

Analysis of Possible Outcomes

To analyze the possible outcomes, we need to consider the possible values of Petya's number after each addition. Let's analyze the possible values of Petya's number after each addition.

1. After the first addition, Petya's number will change to a new value. 2. After the second addition, Petya's number will change again to a new value. 3. This process continues for two hours, with Petya's number changing after each addition.

Since Petya's number can change after each addition, it is not possible to determine the exact values of Petya's number at each step without knowing the specific numbers chosen by Petya and his mother.

Conclusion

Based on the analysis, it is not possible to determine whether Petya's mother can prevent him from going out on that day without knowing the specific numbers chosen by Petya and his mother. The outcome depends on the specific values of Petya's number and his mother's number, as well as the order in which the additions are performed.

Therefore, without additional information, we cannot determine whether Petya's mother can prevent him from going out on that day.