Question

После строительства дома осталось некоторое количество плиток. Их можно использовать для выкладки 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 не полный ряд, в котором на 5 плиток меньше, чем в неполном ряду при укладывании по 7. Сколько всего плиток осталось после строительства дома?

Answer 1

Problem Analysis

We have a certain number of tiles left after building a house, and we want to use them to create a rectangular platform next to the house. When arranging the tiles in rows of 10, we don't have enough tiles to create a square platform. When arranging them in rows of 7, we have one incomplete row left. When arranging them in rows of 8, we also have an incomplete row, but it has 5 fewer tiles than the incomplete row when arranging them in rows of 7. We need to determine the total number of tiles left after building the house.

Solution

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

When arranging the tiles in rows of 10, we don't have 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 perfect square. Let's represent this number as y.

When arranging the tiles in rows of 7, we have one incomplete row left. This means that the number of tiles left after arranging them in rows of 7 is y + 7.

When arranging the tiles in rows of 8, we also have an incomplete row, but it has 5 fewer tiles than the incomplete row when arranging them in rows of 7. This means that the number of tiles left after arranging them in rows of 8 is y + 7 - 5.

Since the number of tiles left after arranging them in rows of 8 is the same as the number of tiles left after arranging them in rows of 7, we can set up the following equation:

y + 7 = y + 7 - 5

Simplifying the equation, we get:

0 = -5

This equation has no solution, which means our assumption that the number of tiles left after building the house is x is incorrect. Therefore, there are no tiles left after building the house.

Answer

After building the house, there are no tiles left.

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