Question

10. Докажите, что числа вида ааа (трехзначные числа, в которых все цифры одинаковы) делятся и на 3 и на 37.​

Answer 1

Пошаговое объяснение:

333:37=9

И когда мы делим на 3

333:3=111

Answer 2

Divisibility of Numbers of the Form "aaa" by 3 and 37

To prove that numbers of the form "aaa" (three-digit numbers in which all the digits are the same) are divisible by both 3 and 37, we can use the properties of divisibility.

Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3. In the case of numbers of the form "aaa," the sum of the digits is 3 times the value of the digit "a." Since all the digits are the same, the sum of the digits is 3a, which is divisible by 3. Therefore, numbers of the form "aaa" are divisible by 3.

Divisibility by 37: To prove that numbers of the form "aaa" are divisible by 37, we need to show that the number can be expressed as 37 multiplied by an integer.

Let's consider an example: 111.

We can express 111 as (100 + 10 + 1), which is equal to (37 * 3 + 37 * 1 + 37 * 1). Therefore, 111 is divisible by 37.

This pattern holds for any number of the form "aaa." We can express it as (100a + 10a + a), which is equal to (37 * 2a + 37 * a + 37 * a). Therefore, any number of the form "aaa" is divisible by 37.

In conclusion, numbers of the form "aaa" are divisible by both 3 and 37.