primenumbers divistion

If P Is a Prime Number Which Divides A2, Then P Divides A, Where A Is a Positive Integer

We know that a number which cannot be written in the form of , where p and q are integers and q ≠ 0, is known as an irrational number.

For example: all numbers of the form, where p is a prime number such asetc., are irrational numbers.

How can we prove that these are irrational numbers?

We can prove this by making use of a theorem which can be stated as follows.

“If p divides a2, then p divides a (where p is a prime number and a is a positive integer)”.

So go through the given video to understand the application of the above stated property.

Similarly, we can prove that square roots of other prime numbers like, etc. are irrational numbers.

Besides these irrational numbers, there are some other irrational numbers like etc.

We can also prove why these numbers are irrational. Before this, let us first see what happens to irrational numbers, when we apply certain mathematical operations on them.

Addition or subtraction of two irrational numbers gives a rational or an irrational number. Addition or subtraction of a rational and an irrational number gives an irrational number. Multiplication of a non-zero rational number and an irrational number gives an irrational number. Multiplication of two irrational numbers gives a rational or an irrational number.

We will now prove that is irrational.

We know that is irrational (as proved before).

Now, the multiplication of a rational and an irrational number gives an irrational number.

Therefore,  is an irrational number.

Let us now try to understand the concept further through some more examples.

Example 1:

Prove that is irrational.

Solution:

Let us assume that is not irrational, i.e. is a rational number.

Then we can write, where a and b are integers andb ≠ 0.

Let a and b have a common factor other than 1.

After dividing by the common factor, we obtain

, where c and d are co-prime numbers.

As c, d and 2 are integers,  and  are rational numbers.

Thus, is rational.

is rational as the difference of two rational numbers is again a rational number.

This is a contradiction as is irrational.

Therefore, our assumption that is rational is wrong.

Hence,  is irrational.

Example 2:

Prove that is irrational.

Solution:

Let us assume  is rational. Then, we can write

,

where a and b are co-prime and b ≠ 0.

⇒

Now, as a and b are integers, is rational or is a rational number.

This means thatis rational.

This is a contradiction as is irrational.

Therefore, our assumption that  is rational is wrong.

Hence,  is an irrational number.