Question

Statement 1 There is no AP with non-zero common difference whose terms are all prime numbers and Statement 2 Any natural number $(>1)$ can be uniquely factored as product of prime numbers.

Name: Problem 111, Chapter 5: IIT-JEE Super Course in Mathematics
Uploaded: 2021-09-09T07:25:25-08:00
Duration: 2 min 32 s
Channel: Gaurav Kalra
Description: Statement 1 There is no AP with non-zero common difference whose terms are all prime numbers and Statement 2 Any natural number (>1) can be uniquely factored as product of prime numbers.

   Statement 1 There is no AP with non-zero common difference whose terms are all prime numbers and
Statement 2 Any natural number $(>1)$ can be uniquely factored as product of prime numbers.

IIT-JEE Super Course in Mathematics

Trishna Knowledge… 1st Edition

Chapter 5, Problem 111

Instant Answer

Solved by Expert Gaurav Kalra

09/09/2021

Step 1

According to the Fundamental Theorem of Arithmetic, any natural number greater than 1 can be uniquely factored as a product of prime numbers. For example, if we take a number like 10, it can be factored into prime numbers as $2 \times 5$. So, Statement 2 is Show more…

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Statement 1 There is no AP with non-zero common difference whose terms are all prime numbers and Statement 2 Any natural number $(>1)$ can be uniquely factored as product of prime numbers.

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Key Concepts

Arithmetic Progression

An arithmetic progression is a sequence of numbers in which the difference between consecutive terms is constant. This concept is crucial in analyzing patterns within sequences, especially when investigating the distribution of particular subsets of numbers, such as primes, along such sequences.

Prime Numbers

Prime numbers are natural numbers greater than 1 that have no divisors other than 1 and themselves. They serve as the fundamental building blocks in number theory and are essential in various proofs and concepts, including the unique factorization of numbers.

Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic states that every natural number greater than 1 can be written uniquely as a product of prime numbers, disregarding the order of the factors. This theorem is central to number theory as it ensures that primes form the basic 'atoms' of the integers.

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