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 6 плиток в ряд остается один неполный ряд, а при укладывании по 7 – тоже остается неполный ряд, в котором на 4 плитки меньше, чем в неполном ряду при укладывании по 8. Сколько всего плиток осталось после строительства дома? Запишите решение и ответ.

Answer 1

Плиток не осталось так как7 *4

Answer 2

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 for a square platform. Arranging them in rows of 6 leaves one incomplete row, and arranging them in rows of 7 also leaves an incomplete row with 4 fewer tiles than the incomplete row when arranging them in rows of 8. 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 for a square platform. This means the number of tiles left is not divisible by 10.

When arranging the tiles in rows of 6, we have one incomplete row. This means the number of tiles left is 1 more than a multiple of 6.

When arranging the tiles in rows of 7, we also have one incomplete row. This means the number of tiles left is 4 less than a multiple of 7.

When arranging the tiles in rows of 8, we have one incomplete row with 4 more tiles than the incomplete row when arranging them in rows of 7. This means the number of tiles left is 4 more than a multiple of 8.

Based on these conditions, we can set up the following equations:

1. x % 10 ≠ 0 (The number of tiles left is not divisible by 10) 2. x % 6 = 1 (The number of tiles left is 1 more than a multiple of 6) 3. x % 7 = 3 (The number of tiles left is 4 less than a multiple of 7) 4. x % 8 = 4 (The number of tiles left is 4 more than a multiple of 8)

To find the value of x, we can use the Chinese Remainder Theorem (CRT) to solve this system of congruences.

Chinese Remainder Theorem (CRT)

The Chinese Remainder Theorem states that if we have a system of congruences of the form:

x ≡ a1 (mod n1)
x ≡ a2 (mod n2)
...
x ≡ ak (mod nk)

where a1, a2, ..., ak are remainders and n1, n2, ..., nk are pairwise coprime (i.e., they have no common factors), then there exists a unique solution for x modulo N, where N = n1 * n2 * ... * nk.

In our case, we have the following congruences:

1. x ≡ 0 (mod 10) 2. x ≡ 1 (mod 6) 3. x ≡ 3 (mod 7) 4. x ≡ 4 (mod 8)

To find the value of x, we can apply the Chinese Remainder Theorem.

Applying the Chinese Remainder Theorem

Let's find the value of x using the Chinese Remainder Theorem.

We can start by finding the value of N:

N = 10 * 6 * 7 * 8 = 3360

Now, let's find the values of Ni for each congruence:

1. Ni = N / 10 = 336 2. Ni = N / 6 = 560 3. Ni = N / 7 = 480 4. Ni = N / 8 = 420

Next, let's find the values of yi for each congruence, where yi is the modular inverse of Ni modulo ni:

1. yi ≡ 6 (mod 10) 2. yi ≡ 5 (mod 6) 3. yi ≡ 1 (mod 7) 4. yi ≡ 5 (mod 8)

Finally, we can calculate the value of x using the formula:

x = (a1 * N1 * y1 + a2 * N2 * y2 + a3 * N3 * y3 + a4 * N4 * y4) % N

Substituting the values:

x = (0 * 336 * 6 + 1 * 560 * 5 + 3 * 480 * 1 + 4 * 420 * 5) % 3360

Calculating this expression, we find that x = 2016.

Therefore, the total number of tiles left after building the house is 2016.

Answer

The total number of tiles left after building the house is 2016.