A number is multiplied by 11 and 11 is added to the product. If the resulting number is divisible by 13, what is the smallest number?53262212

The problem can be solved by using algebra. Let's denote the unknown number as x. According to the problem, the number is multiplied by 11 and then 11 is added to the product. This can be written as:

11x + 11

The problem also states that the resulting number is divisible by 13. This can be written as:

11x + 11 = 13y

where y is some integer.

To find the smallest possible value for x, we can start by setting y to 1 and solve for x:

11x + 11 = 13 11x = 13 - 11 11x = 2 x = 2/11

Since x must be an integer, 2/11 is not a valid solution. We can then try with y = 2:

11x + 11 = 26 11x = 26 - 11 11x = 15 x = 15/11

Again, x must be an integer, so 15/11 is not a valid solution. We can continue this process until we find an integer solution for x. If we try with y = 11:

11x + 11 = 143 11x = 143 - 11 11x = 132 x = 132/11 x = 12

So, the smallest number that satisfies the conditions of the problem is 12.