Question

Сколько различных натуральных чисел от 4 до 1004 включительно, не делящихся ни на 5, ни на 8?

Answer 1

И на 5 и на 8 делятся все числа кратные 5*8 = 40. Следовательно не делятся на них все остальные.
Наибольшее кратное 40 - 1000 
Чисел кратных 40 - 1000/40 = 25
Всего чисел 1004-3=1001
Тогда чисел не делящихся ни на 5 ни на 8 1001-25=976
Ответ: 976. 

 

Answer 2

Calculation of the Number of Natural Numbers

To determine the number of natural numbers between 4 and 1004 (inclusive) that are not divisible by 5 or 8, we can use the principle of inclusion-exclusion.

First, let's calculate the number of natural numbers between 4 and 1004 (inclusive) that are divisible by 5. We can use the formula for the number of terms in an arithmetic progression:

Number of terms = (last term - first term) / common difference + 1

In this case, the first term is 5, the last term is 1000 (the largest multiple of 5 less than or equal to 1004), and the common difference is 5. Plugging these values into the formula, we get:

Number of terms divisible by 5 = (1000 - 5) / 5 + 1 = 200

Next, let's calculate the number of natural numbers between 4 and 1004 (inclusive) that are divisible by 8. Using the same formula, we have:

Number of terms divisible by 8 = (1000 - 8) / 8 + 1 = 125

However, we need to consider the numbers that are divisible by both 5 and 8. These numbers are multiples of the least common multiple (LCM) of 5 and 8, which is 40. We can calculate the number of terms divisible by 40 using the same formula:

Number of terms divisible by 40 = (1000 - 40) / 40 + 1 = 25

Now, we can use the principle of inclusion-exclusion to calculate the number of natural numbers between 4 and 1004 (inclusive) that are not divisible by 5 or 8. The formula is:

Number of terms not divisible by 5 or 8 = Total number of terms - Number of terms divisible by 5 - Number of terms divisible by 8 + Number of terms divisible by 40

Plugging in the values we calculated earlier, we get:

Number of terms not divisible by 5 or 8 = 1001 - 200 - 125 + 25 = 701

Therefore, there are 701 different natural numbers between 4 and 1004 (inclusive) that are not divisible by 5 or 8.