Question

If n is any prime number and a2 is divisible by n, then n will also divide a. Justify.

Answer 1

Step-by-step explanation:

Given :-

n is a prime number and a² is divisible by n.

To find :-

Prove that n is also divides a ?

Solution :-

Given that :

n is any prime number.

a is a positive integer.

n divides a²-------(1)

If n divides a² then a² is a multiple of n

a² is in the form of prime factors

Let n1,n2,...,nm be the factors of a

=> a = n1.n2.n3...are prime numbers not necessarily all distinct.

Therefore, a² = a×a

=> a² = (n1.n2.n3...)(n1.n2.n3...)

=> a² = (n1)².(n2)².(n3)²...(nm)²

=> n is a prime factor of a²

From Fundamental Theorem of Arithmetic,

The factors of a² are only n1,n2,...nm----(2)

(Uniqueness of prime factors )

From (1)&(2)

We have,

n is a prime factors of n1,n2,..nm

=> n divides a

=> a is divided by a.

Hence , Proved.

Answer:-

If n is any prime number and a2 is divisible by n, then n will also divide a.

Used formulae:-

Fundamental Theorem of Arithmetic

Every composite number can be expressed as the product of prime factors and this prime factorization is unique apart from the order in which the prime factors occur.

Example :-

Let 2 be a prime number divides 6² then it divides 6 also

6²/2 = 36/2 = 18

6/2 = 3

Answer 2

Answer:

n -4 is divided into two

Step-by-step explanation:

first

2,4,6,8 etc primer no and this no is divided into other no