Question

Корзина полная яблок,вмещает не более 500 яблок.Если бы их вынимали по 2,по 3,по 4,по 5 или по 6,то осталось бы 1 яблоко.Их вынимали по 7,и остатка не получилось.Сколько яблок было в корзине?

Answer 1

Х+1-число яблок в корзине
Число х должно быть кратно 2,3,4,5,6 
Значит должно быть четным,сумма цифр делиться на 3,две последние цифры делиться на 4,оканчиваться 0
Это числа кратные 60 и меньше 500
х=480
Значит в корзине 481 яблоко
481:2=240(1)
481:3=160(1)
481:4=120(1)
481:5=96(1)
481:6=80(1)
481:7=68(5)

Answer 2

Problem Analysis

The problem states that a basket is full of apples and can hold no more than 500 apples. If the apples are taken out in groups of 2, 3, 4, 5, or 6, there would be 1 apple left. However, when the apples are taken out in groups of 7, there is no remainder. The goal is to determine how many apples were originally in the basket.

Solution

Let's assume that there were x apples in the basket initially.

If we take out the apples in groups of 2, 3, 4, 5, or 6, there would be 1 apple left. This can be represented by the following equations:

- x ≡ 1 (mod 2) - x ≡ 1 (mod 3) - x ≡ 1 (mod 4) - x ≡ 1 (mod 5) - x ≡ 1 (mod 6)

If we take out the apples in groups of 7, there is no remainder. This can be represented by the equation:

- x ≡ 0 (mod 7)

To find the value of x, we can solve this system of congruences.

Solution Steps

1. Start with the equation x ≡ 1 (mod 2). 2. Increment x by 2 until it satisfies the equation x ≡ 1 (mod 3). 3. Increment x by 6 until it satisfies the equation x ≡ 1 (mod 4). 4. Increment x by 4 until it satisfies the equation x ≡ 1 (mod 5). 5. Increment x by 4 until it satisfies the equation x ≡ 1 (mod 6). 6. Increment x by 1 until it satisfies the equation x ≡ 0 (mod 7).

Let's calculate the value of x using these steps.

Calculation

Starting with x = 1, let's increment x by the required values until it satisfies all the congruence equations:

- x = 1 (mod 2) - x = 3 (mod 3) - x = 7 (mod 4) - x = 11 (mod 5) - x = 15 (mod 6) - x = 15 (mod 7)

The smallest positive value of x that satisfies all the congruence equations is x = 15.

Therefore, there were 15 apples in the basket initially.

Conclusion

The basket initially contained 15 apples.