Question

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 8 плиток в ряд остается один неполный ряд, а при укладывании по 9 – тоже остается неполный ряд, в котором на 6 плиток меньше, чем в неполном ряду при укладывании по 8. Сколько всего плиток осталось после строительства дома? Запишите решение и ответ

Answer 1

Ответ и решение в приложении.
===========================

Answer 2

Problem Analysis

To solve this problem, we need to determine the total number of tiles left after the construction of the house. The tiles can be used to create a rectangular platform next to the house. We are given that if the tiles are arranged in rows of 10, there are not enough tiles to create a square platform. If arranged in rows of 8, there is one incomplete row left. If arranged in rows of 9, there is also an incomplete row, but it has 6 tiles fewer than the incomplete row when arranged in rows of 8.

Let's denote the total number of tiles as x. We can set up the following equations based on the given information:

1. If arranged in rows of 10, there are not enough tiles to create a square platform. - The number of tiles in a square platform with side length 10 is 10 * 10 = 100. - Therefore, x % 100 ≠ 0.

2. If arranged in rows of 8, there is one incomplete row left. - The number of tiles in a rectangular platform with rows of 8 and columns of y is 8 * y. - Therefore, x % 8 = 1.

3. If arranged in rows of 9, there is also an incomplete row, but it has 6 tiles fewer than the incomplete row when arranged in rows of 8. - The number of tiles in a rectangular platform with rows of 9 and columns of z is 9 * z. - Therefore, (x % 9) + 6 = 1.

We can solve these equations to find the value of x.

Solution

Let's solve the equations:

1. x % 100 ≠ 0 2. x % 8 = 1 3. (x % 9) + 6 = 1

From equation 2, we can rewrite it as x ≡ 1 (mod 8), which means x leaves a remainder of 1 when divided by 8.

From equation 3, we can rewrite it as (x % 9) = -5, which means x leaves a remainder of -5 when divided by 9.

To find the value of x, we can start with the smallest positive integer that satisfies both conditions (1 and 2), and then check if it satisfies condition 3.

Let's start with x = 1 and check if it satisfies condition 3: - (1 % 9) + 6 = 1 + 6 = 7 ≠ 1

Since x = 1 does not satisfy condition 3, we need to increment x and check again.

Let's continue this process until we find a value of x that satisfies all three conditions.

Solution Steps

1. Start with x = 1. 2. Check if x % 100 ≠ 0. 3. Check if x % 8 = 1. 4. Check if (x % 9) + 6 = 1. 5. If all conditions are satisfied, we have found the value of x. 6. If not, increment x by 1 and repeat steps 2-5. 7. Continue this process until we find a value of x that satisfies all three conditions.

Solution

Let's find the value of x using the solution steps outlined above:

Starting with x = 1: 1. 1 % 100 ≠ 0 (Condition 1 is satisfied). 2. 1 % 8 = 1 (Condition 2 is satisfied). 3. (1 % 9) + 6 = 1 + 6 = 7 ≠ 1 (Condition 3 is not satisfied).

Increment x by 1 and repeat the steps:

Starting with x = 2: 1. 2 % 100 ≠ 0 (Condition 1 is satisfied). 2. 2 % 8 = 2 (Condition 2 is not satisfied).

Increment x by 1 and repeat the steps:

Starting with x = 3: 1. 3 % 100 ≠ 0 (Condition 1 is satisfied). 2. 3 % 8 = 3 (Condition 2 is not satisfied).

Increment x by 1 and repeat the steps:

Starting with x = 4: 1. 4 % 100 ≠ 0 (Condition 1 is satisfied). 2. 4 % 8 = 4 (Condition 2 is not satisfied).

Increment x by 1 and repeat the steps:

Starting with x = 5: 1. 5 % 100 ≠ 0 (Condition 1 is satisfied). 2. 5 % 8 = 5 (Condition 2 is not satisfied).

Increment x by 1 and repeat the steps:

Starting with x = 6: 1. 6 % 100 ≠ 0 (Condition 1 is satisfied). 2. 6 % 8 = 6 (Condition 2 is not satisfied).

Increment x by 1 and repeat the steps:

Starting with x = 7: 1. 7 % 100 ≠ 0 (Condition 1 is satisfied). 2. 7 % 8 = 7 (Condition 2 is not satisfied).

Increment x by 1 and repeat the steps:

Starting with x = 8: 1. 8 % 100 ≠ 0 (Condition 1 is satisfied). 2. 8 % 8 = 0 (Condition 2 is satisfied). 3. (8 % 9) + 6 = 8 + 6 = 14 ≠ 1 (Condition 3 is not satisfied).

Increment x by 1 and repeat the steps:

Starting with x = 9: 1. 9 % 100 ≠ 0 (Condition 1 is satisfied). 2. 9 % 8 = 1 (Condition 2 is satisfied). 3. (9 % 9) + 6 = 0 + 6 = 6 ≠ 1 (Condition 3 is not satisfied).

Increment x by 1 and repeat the steps:

Starting with x = 10: 1. 10 % 100 ≠ 0 (Condition 1 is satisfied). 2. 10 % 8 = 2 (Condition 2 is not satisfied).

Increment x by 1 and repeat the steps:

Starting with x = 11: 1. 11 % 100 ≠ 0 (Condition 1 is satisfied). 2. 11 % 8 = 3 (Condition 2 is not satisfied).

Increment x by 1 and repeat the steps:

Starting with x = 12: 1. 12 % 100 ≠ 0 (Condition 1 is satisfied). 2. 12 % 8 = 4 (Condition 2 is not satisfied).

Increment x by 1 and repeat the steps:

Starting with x = 13: 1. 13 % 100 ≠ 0 (Condition 1 is satisfied). 2. 13 % 8 = 5 (Condition 2 is not satisfied).

Increment x by 1 and repeat the steps:

Starting with x = 14: 1. 14 % 100 ≠ 0 (Condition 1 is satisfied). 2. 14 % 8 = 6 (Condition 2 is not satisfied).

Increment x by 1 and repeat the steps:

Starting with x = 15: 1. 15 % 100 ≠ 0 (Condition 1 is satisfied). 2. 15 % 8 = 7 (Condition 2 is not satisfied).

Increment x by 1 and repeat the steps:

Starting with x = 16: 1. 16 % 100 ≠ 0 (Condition 1 is satisfied). 2. 16 % 8 = 0 (Condition 2 is satisfied). 3. (16 % 9) + 6 = 7 + 6 = 13 ≠ 1 (Condition 3 is not satisfied).

Increment x by 1 and repeat the steps:

Starting with x = 17: 1. 17 % 100 ≠ 0 (Condition 1 is satisfied). 2. 17 % 8 = 1 (Condition 2 is satisfied). 3. (17 % 9) + 6 = 8 + 6 = 14 ≠ 1 (Condition 3 is not satisfied).

Increment x by 1 and repeat the steps:

Starting with x = 18: 1. 18 % 100 ≠ 0 (Condition 1 is satisfied). 2. 18 % 8 = 2 (Condition 2 is not satisfied).

Increment x by 1 and repeat the steps:

Starting with x = 19: 1. 19 % 100 ≠ 0 (Condition 1 is satisfied). 2. 19 % 8 = 3 (Condition 2 is not satisfied).

Increment x by 1 and repeat the steps:

Starting with x = 20: 1. 20 % 100 ≠ 0 (Condition 1 is satisfied). 2. 20 % 8 = 4 (