Question

Theorem 1.3 in chapter 1 . REAL NUMBERS class 10
Explain clearly with proof
I want clear explaination
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Answer 1

It means that if a number can be divide from any other number's square then it will also divide by that number

Answer 2

Answer:

Theorem:1.3

Let p be a prime number. If p divides a^2, then p divides a, where a is a positive integer.

Proof:

Let the prime factorisation of a as follows:

Let the prime factorisation of a as follows:a=p1p2..........pn, where p1,p2,..........,pn are primes, not necessarily distinct.

Let the prime factorisation of a as follows:a=p1p2..........pn, where p1,p2,..........,pn are primes, not necessarily distinct.Therefore, a^2 = (p1p2.........pn) (p1p2.........pn)=p1^2 p2^2............pn^2.

Let the prime factorisation of a as follows:a=p1p2..........pn, where p1,p2,..........,pn are primes, not necessarily distinct.Therefore, a^2 = (p1p2.........pn) (p1p2.........pn)=p1^2 p2^2............pn^2.Now, we are given that p divides a^2.

Let the prime factorisation of a as follows:a=p1p2..........pn, where p1,p2,..........,pn are primes, not necessarily distinct.Therefore, a^2 = (p1p2.........pn) (p1p2.........pn)=p1^2 p2^2............pn^2.Now, we are given that p divides a^2. Therefore, from the Fundamental Theorem of Arithmetic , it follows that p is one of the prime factors of a^2.

Let the prime factorisation of a as follows:a=p1p2..........pn, where p1,p2,..........,pn are primes, not necessarily distinct.Therefore, a^2 = (p1p2.........pn) (p1p2.........pn)=p1^2 p2^2............pn^2.Now, we are given that p divides a^2. Therefore, from the Fundamental Theorem of Arithmetic , it follows that p is one of the prime factors of a^2.However, using uniqueness part of the Fundamental Theorem of Arithmetic , we realise that the only prime factors of a^2 are p1, p2,....pn.

Let the prime factorisation of a as follows:a=p1p2..........pn, where p1,p2,..........,pn are primes, not necessarily distinct.Therefore, a^2 = (p1p2.........pn) (p1p2.........pn)=p1^2 p2^2............pn^2.Now, we are given that p divides a^2. Therefore, from the Fundamental Theorem of Arithmetic , it follows that p is one of the prime factors of a^2.However, using uniqueness part of the Fundamental Theorem of Arithmetic , we realise that the only prime factors of a^2 are p1, p2,....pn.So, p is one of p1, p2,......pn.

Let the prime factorisation of a as follows:a=p1p2..........pn, where p1,p2,..........,pn are primes, not necessarily distinct.Therefore, a^2 = (p1p2.........pn) (p1p2.........pn)=p1^2 p2^2............pn^2.Now, we are given that p divides a^2. Therefore, from the Fundamental Theorem of Arithmetic , it follows that p is one of the prime factors of a^2.However, using uniqueness part of the Fundamental Theorem of Arithmetic , we realise that the only prime factors of a^2 are p1, p2,....pn.So, p is one of p1, p2,......pn.Now, since a=p1p2.......pn, p divides a.

Let the prime factorisation of a as follows:a=p1p2..........pn, where p1,p2,..........,pn are primes, not necessarily distinct.Therefore, a^2 = (p1p2.........pn) (p1p2.........pn)=p1^2 p2^2............pn^2.Now, we are given that p divides a^2. Therefore, from the Fundamental Theorem of Arithmetic , it follows that p is one of the prime factors of a^2.However, using uniqueness part of the Fundamental Theorem of Arithmetic , we realise that the only prime factors of a^2 are p1, p2,....pn.So, p is one of p1, p2,......pn.Now, since a=p1p2.......pn, p divides a.Hence,proved.

Let the prime factorisation of a as follows:a=p1p2..........pn, where p1,p2,..........,pn are primes, not necessarily distinct.Therefore, a^2 = (p1p2.........pn) (p1p2.........pn)=p1^2 p2^2............pn^2.Now, we are given that p divides a^2. Therefore, from the Fundamental Theorem of Arithmetic , it follows that p is one of the prime factors of a^2.However, using uniqueness part of the Fundamental Theorem of Arithmetic , we realise that the only prime factors of a^2 are p1, p2,....pn.So, p is one of p1, p2,......pn.Now, since a=p1p2.......pn, p divides a.Hence,proved.The proof is based on a technique called 'proof by contradiction'.