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Divisibility is a central concept in understanding numbers. It refers to the ability of a number to be divided by another number without leaving a remainder. For instance, 10 is divisible by 2 because when 10 is divided by 2, the result is an integer, 5, without any remainder.

In the context of prime numbers, divisibility is crucial. A prime number is defined as one that can only be divided evenly by 1 and itself. To check if a number, like the ones in our problem, is prime, we examine if it has any divisors other than 1 and the number itself.

For the sequence given by the formula \( t_n = \frac{10^n - 1}{9} \), determining primality involves checking if the resulting number is divisible by smaller primes. For instance:

• If \( 10^n - 1 \) is divisible by any factor other than itself, then \( t_n \) is not prime.
• Understanding divisibility helps identify properties such as composite numbers which have additional divisors.

Composite numbers are the opposite of prime numbers. A composite number is any integer greater than one that is not prime, meaning it has divisors other than 1 and itself.

To identify composite numbers we look for number representations that allow factors and divisors.

For our task, where each number \( t_n \) is represented by repeating '1' for \( n \) times, checking divisibility by smaller primes, like 3 or 5, helps to quickly determine composite nature.

• If \( n \) is divisible by a smaller number like 3, the resulting \( 10^n - 1 \) often becomes divisible too. For example, 912 and 951 being divisible by 3 suggests their composite status.
• Composite numbers reveal themselves when the formula \( \frac{10^n - 1}{9} \) yields non-prime through factorization.

Number representation refers to how numbers are expressed or formatted. It plays a vital role in problems dealing with sequences and patterns.

In our problem, the number \( t_n = \underbrace{111\ldots1}_{n\text{ times}} \) is efficiently represented by the formula \( t_n = \frac{10^n - 1}{9} \). This concise representation aids in exploring its properties, such as divisibility and primality.

• The representation \( \frac{10^n - 1}{9} \) splits a repetitive number into a numeric expression, simplifying complex calculations.
• This representation uncovers the structure and factors of numbers, allowing easier evaluation of their property, whether prime or composite.
• Understanding the representation aids in tackling diverse mathematical problems efficiently.