Question

Когда завершилось строительство офисного помещения, у рабочих осталось некоторое число плиток. Было решено выложить прямоугольную площадку рядом со зданием. Если укладывать по 10 плиток в ряд, то на последний ряд плитки не хватит. Если укладывать по 11 плиток в ряду, то полных рядов получится столько же, но в последнем ряду плиток будет на 8 плиток(-ок, -и) меньше, чем в последнем ряду, когда укладываешь по 10. В итоге рабочие уложили по 12 плиток в одном ряду, осталось несколько лишних плиток, и квадратную площадку сделать не получилось. Сколько плиток было изначально?

Answer 1

Ответ:

89

Пошаговое объяснение:

Answer 2

Problem Analysis

The problem states that after the construction of an office building, there were some leftover tiles. The workers decided to create a rectangular platform next to the building using these tiles. If they lay 10 tiles in a row, there would be not enough tiles for the last row. If they lay 11 tiles in a row, there would be the same number of full rows, but the last row would be 8 tiles shorter than the last row when laying 10 tiles. In the end, the workers laid 12 tiles in each row, but there were still some extra tiles left, and they couldn't create a square platform. We need to determine the initial number of tiles.

Solution

Let's assume the initial number of tiles is x.

If the workers lay 10 tiles in a row, there will be x % 10 tiles left.

If the workers lay 11 tiles in a row, there will be x % 11 tiles left, and the last row will be 8 tiles shorter than the last row when laying 10 tiles.

If the workers lay 12 tiles in a row, there will be x % 12 tiles left.

Since there were still some extra tiles left, we can set up the following equation:

x % 10 = x % 11 = x % 12

To solve this equation, we can find the least common multiple (LCM) of 10, 11, and 12, and add it to the equation:

x % 10 = x % 11 = x % 12 = LCM(10, 11, 12)

Let's calculate the LCM of 10, 11, and 12:

LCM(10, 11, 12) = 660

Now we can solve the equation:

x % 10 = x % 11 = x % 12 = 660

The initial number of tiles, x, is equal to 660.

Answer

The initial number of tiles was 660.