Final answer:
A 12-digit number formed by repeating a three-digit number is exactly divisible by 13, 37, and 111, because it can be represented as the product of the repeated three-digit number and the factor 1,001,001, which is divisible by these numbers.
Step-by-step explanation:
A 12-digit number formed by repeating a three-digit number, such as 978978978978, will always be exactly divisible by certain specific numbers. This type of number is divisible by 13, 37, and 111.
The reason for this is based on the fact that when you have a repeating number pattern like this, it can be broken down into a product of the repeated number and a specific factor.
For example, the number 978978978978 is equivalent to 978 multiplied by 1,001,001, where 1,001,001 is a factor that results from 10,000,000,000 + 1,000,000 + 1, or more simply put, 108 + 104 + 1.
This factor, when broken down further, is divisible by 13, 37, and 111, which is why any number of the given form will be divisible by these numbers.