Problem 115 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(t_{n}=\underbrace{1.1 \ldots 1}_{n \text { times }}\), then (A) \(t_{912}\) is not prime (B) \(t_{951}\) is not prime (C) \(t_{480}\) is not prime (D) \(t_{91}\) is not prime

Step-by-Step Solution

Answer

(D) \( t_{91} \) is not prime.

Step 1: Understanding the Problem

We are given a sequence where the term \( t_n \) is represented by the number 111...1, repeated \( n \) times. We need to determine the primality of the number for the given choices. Each option corresponds to a different \( n \): 912, 951, 480, and 91.

Step 2: Checking Prime Number Definition

A prime number is only divisible by 1 and itself. To check if \( t_n \) is prime, we need to see if there are any divisors other than 1 and \( t_n \).

Step 3: Converting Numbers to Formula

The number \( t_n \) can be expressed as \( t_n = \frac{10^n - 1}{9} \). This formula helps in analyzing its divisibility properties.

Step 4: Checking Divisibility for Each Option

Since \( t_n = \frac{10^n - 1}{9} \), we check if \( 10^n - 1 \) is divisible by \( 9k \) for some integer \( k \), hence making \( t_n \) composite. We specifically consider divisibility by smaller primes or forms like \( 10^k - 1 \) that factorize it.

Step 5: Analyzing Options

1. For \( t_{912} \), notice that 912 is divisible by 3 (since 9+1+2=12, and 12 is divisible by 3), and thus \( 10^{912} - 1 \) will be divisible by \( 10^3 - 1 = 999 \), making \( t_{912} \) composite.2. For \( t_{951} \), 951 is divisible by 3 (9+5+1=15, and 15 is divisible by 3), so \( 10^{951} - 1 \) is divisible by \( 10^3 - 1 \).3. For \( t_{480} \), 480 is divisible by 3 (4+8+0=12, and 12 is divisible by 3), repeating the reasoning as before.4. Check divisibility of \( 10^{91} - 1 \) by simpler factors (consider divisibility by the form \( 10^p - 1 \) for any smaller \( p \), e.g., using Mersenne-like factorization if any).

Step 6: Identifying the Prime Option

For exactly one of these cases (\( t_{n} \)), the number turns out not to have such factorization properties suitable for making it non-prime. By identifying the outlier among \(91, 912, 951, 480\), we find that \( t_{91} \) doesn't immediately present obvious small divisibility, hinting toward it possibly being a prime number.

Key Concepts

Divisibility

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

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

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.