Question

14. После строительства дома осталось некоторое количество плиток. Их можно использовать для выкладывания прямоугольной площадки на участке рядом с домом. Если укладывать в ряд по 10 плиток, то для квадратной площадки плиток не хватает. При укладывании по 7 плиток в ряд остается один неполный ряд, а при укладывании по 8 – тоже остается неполный ряд, в котором на 5 плиток меньше, чем в неполном ряду при укладывании по 7. Сколько всего плиток осталось после строительства дома? Запишите решение и ответ.

Answer 1

Решение в приложении.
Условию задачи удовлетворяют два ответа:
1) 41 плитка
2) 97  плиток.

Answer 2

Problem Analysis

We are given that there are some leftover tiles after constructing a house. These tiles can be used to create a rectangular platform next to the house. When arranging the tiles in rows of 10, there are not enough tiles to create a square platform. Similarly, when arranging the tiles in rows of 7 or 8, there are incomplete rows with a difference of 5 tiles. We need to determine the total number of tiles that are left after constructing the house.

Solution

Let's assume the total number of tiles left after constructing the house is x.

According to the given information, when arranging the tiles in rows of 10, there are not enough tiles to create a square platform. This means that the number of tiles left after arranging them in rows of 10 is not a multiple of 10. Therefore, we can write the equation:

x ≠ 10a (Equation 1)

where a is an integer.

Similarly, when arranging the tiles in rows of 7, there is one incomplete row, and when arranging them in rows of 8, there is also one incomplete row with 5 fewer tiles than the incomplete row when arranging them in rows of 7. This can be expressed as:

x = 7b + 1 (Equation 2)

x = 8c + 5 (Equation 3)

where b and c are integers.

To find the value of x, we can solve the system of equations formed by Equations 1, 2, and 3.

Solving the System of Equations

Let's solve the system of equations using substitution.

From Equation 2, we can express b in terms of x:

b = (x - 1) / 7 (Equation 4)

Substituting Equation 4 into Equation 3, we get:

x = 8c + 5 = 8((x - 1) / 7) + 5

Simplifying the equation:

7x = 56c + 40

Dividing both sides by 7:

x = 8c + 5.71

Since x is an integer, 8c + 5.71 must be an integer. This means that c must be a multiple of 7. Let's substitute c = 7d into the equation:

x = 8(7d) + 5.71

Simplifying the equation:

x = 56d + 5.71

Since x is an integer, 56d + 5.71 must be an integer. This means that d must be a multiple of 100. Let's substitute d = 100e into the equation:

x = 56(100e) + 5.71

Simplifying the equation:

x = 5600e + 5.71

Since x is an integer, 5600e + 5.71 must be an integer. This means that e must be 0. Therefore, d = 100e = 100(0) = 0.

Substituting d = 0 into the equation:

x = 5600(0) + 5.71

Simplifying the equation:

x = 5.71

Since x is an integer, the solution is not valid. This means that there is no solution that satisfies all the given conditions.

Conclusion

Based on the given information, there is no solution for the total number of tiles left after constructing the house.