Question

Вася изменяет число, написанное на доске, по следующему правилу: если это число делится на 3, то Вася вычитает из него 1. Если при делени на 3 число даёт остаток 2, то Вася вычитает из него 2, а если число при делении на 3 даёт остаток 1, то Вася прибавляет к нему 2. Он начинает с числа 160, какое число получит Вася после 28 таких операций? СРОЧНО ПОМОГИТЕ!!!!!!!!!!

Answer 1

Первые 2 операции составляют цикл уменьшения числа на 3:
число 3k⇒3k-1=3(k-1)+2⇒3(k-1)
Последняя операция превращает число 3k+1 в 3k+3 - то есть, может быть осуществлена только 1 раз, далее переходи к циклу.
Так как 160=53*3+1, то после первой операции оно превратится в 162=54*3.
Далее 26 операций уменьшат его на 3*(26/2)=49. То есть, получится 162-49=113.
Далее, 28-я операция вычтет из этого числа 1 и получится 112.
Ответ: 112

Answer 2

Problem Analysis

In this problem, Vasya is given a number written on a board and follows a set of rules to modify it. If the number is divisible by 3, Vasya subtracts 1 from it. If the number gives a remainder of 2 when divided by 3, Vasya subtracts 2 from it. And if the number gives a remainder of 1 when divided by 3, Vasya adds 2 to it. Vasya starts with the number 160 and performs this operation 28 times. We need to determine the number that Vasya will have after these 28 operations.

Solution

To solve this problem, we can simulate the operations performed by Vasya on the number 160. We will keep track of the number after each operation and repeat the process 28 times.

Starting with the number 160: - Since 160 is divisible by 3, Vasya subtracts 1 from it and gets 159. - 159 gives a remainder of 0 when divided by 3, so Vasya subtracts 1 from it and gets 158. - 158 gives a remainder of 2 when divided by 3, so Vasya subtracts 2 from it and gets 156. - 156 gives a remainder of 0 when divided by 3, so Vasya subtracts 1 from it and gets 155. - 155 gives a remainder of 2 when divided by 3, so Vasya subtracts 2 from it and gets 153. - 153 gives a remainder of 0 when divided by 3, so Vasya subtracts 1 from it and gets 152. - 152 gives a remainder of 2 when divided by 3, so Vasya subtracts 2 from it and gets 150. - 150 gives a remainder of 0 when divided by 3, so Vasya subtracts 1 from it and gets 149. - 149 gives a remainder of 2 when divided by 3, so Vasya subtracts 2 from it and gets 147. - 147 gives a remainder of 0 when divided by 3, so Vasya subtracts 1 from it and gets 146. - 146 gives a remainder of 1 when divided by 3, so Vasya adds 2 to it and gets 148. - 148 gives a remainder of 1 when divided by 3, so Vasya adds 2 to it and gets 150. - 150 gives a remainder of 0 when divided by 3, so Vasya subtracts 1 from it and gets 149. - 149 gives a remainder of 2 when divided by 3, so Vasya subtracts 2 from it and gets 147. - 147 gives a remainder of 0 when divided by 3, so Vasya subtracts 1 from it and gets 146. - 146 gives a remainder of 1 when divided by 3, so Vasya adds 2 to it and gets 148. - 148 gives a remainder of 1 when divided by 3, so Vasya adds 2 to it and gets 150. - 150 gives a remainder of 0 when divided by 3, so Vasya subtracts 1 from it and gets 149. - 149 gives a remainder of 2 when divided by 3, so Vasya subtracts 2 from it and gets 147. - 147 gives a remainder of 0 when divided by 3, so Vasya subtracts 1 from it and gets 146. - 146 gives a remainder of 1 when divided by 3, so Vasya adds 2 to it and gets 148. - 148 gives a remainder of 1 when divided by 3, so Vasya adds 2 to it and gets 150. - 150 gives a remainder of 0 when divided by 3, so Vasya subtracts 1 from it and gets 149. - 149 gives a remainder of 2 when divided by 3, so Vasya subtracts 2 from it and gets 147. - 147 gives a remainder of 0 when divided by 3, so Vasya subtracts 1 from it and gets 146. - 146 gives a remainder of 1 when divided by 3, so Vasya adds 2 to it and gets 148. - 148 gives a remainder of 1 when divided by 3, so Vasya adds 2 to it and gets 150. - 150 gives a remainder of 0 when divided by 3, so Vasya subtracts 1 from it and gets 149. - 149 gives a remainder of 2 when divided by 3, so Vasya subtracts 2 from it and gets 147. - 147 gives a remainder of 0 when divided by 3, so Vasya subtracts 1 from it and gets 146.

After 28 operations, Vasya will have the number 146.

Answer

After 28 operations, Vasya will have the number 146.