Question

Если имеющиеся карандаши разложить в коробки , по 8 штук в каждую , то останется 5 лишних карандашей. Если их разложить в каробки, по 6 штук в каждую , то тоже останется 5 лишних карандашей . Сколько имеется карандашей , если их больше 50,но меньше 100?Ответ ___ или ___

Answer 1

Ответ:

53 или 77 карандашей

Пошаговое объяснение:

Answer 2

Problem Analysis

We are given that if we distribute the available pencils into boxes of 8, there will be 5 extra pencils. Similarly, if we distribute them into boxes of 6, there will also be 5 extra pencils. We need to determine the number of pencils, which is greater than 50 but less than 100, that satisfy these conditions.

Solution

Let's assume the number of pencils is x.

According to the given information, we can set up the following equations:

Equation 1: x ≡ 5 (mod 8) . Equation 2: x ≡ 5 (mod 6) .

To find the solution, we can solve these congruences simultaneously.

Chinese Remainder Theorem

The Chinese Remainder Theorem (CRT) 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 n1, n2, ..., nk are pairwise coprime (i.e., they have no common factors), then there exists a unique solution modulo N, where N = n1 * n2 * ... * nk.

In our case, we have: x ≡ 5 (mod 8) x ≡ 5 (mod 6)

Since 8 and 6 are not coprime (they share a common factor of 2), we need to find a different approach to solve this system of congruences.

Extended Euclidean Algorithm

To find a solution, we can use the Extended Euclidean Algorithm to find the modular inverse of one of the moduli.

In our case, we can find the modular inverse of 8 modulo 6.

Using the Extended Euclidean Algorithm, we find that the modular inverse of 8 modulo 6 is 5.

Applying the Modular Inverse

Now that we have the modular inverse, we can rewrite Equation 1 as: x ≡ 5 * 8 * (8^(-1) mod 6) + 5 * 6 * (6^(-1) mod 8) (mod 48).

Simplifying this equation, we get: x ≡ 40 + 30 (mod 48) x ≡ 70 (mod 48).

Therefore, the solution to the system of congruences is x ≡ 70 (mod 48).

Finding the Range of x

We are given that x is greater than 50 but less than 100.

To find the possible values of x within this range, we can add or subtract multiples of 48 from 70 until we find a value that satisfies the condition.

By subtracting 48 from 70, we get: x ≡ 22 (mod 48).

The possible values of x within the given range are 22, 70, and 118.

However, we need to choose the value of x that is greater than 50 but less than 100. Therefore, the only valid solution is x = 70.

Answer

The number of pencils is 70.

Note: The solution is based on the assumption that the number of pencils is an integer.