Answer

Verified

403.8k+ views

**Hint:**In this question, we will use the contradiction method to prove the given statement. First, we will assume that the square of a number is equal to 3n-1. After this, we will use Fermat’s little theorem to contradict the assumption. We will also see what Fermat’s little theorem is?

**Complete step-by-step solution:**We will first assume a number N.

And let ${N^2}= 3n-1.$

$ \Rightarrow $ ${N^2} +1 = 3n.$

It is clear from the above equation that ${N^2}+1$ is the multiple of 3.

Now, we will rearrange the term ${N^2}+1$ by adding and subtracting 1 as follow:

${N^2}$+1= ${N^2}+1+1-1 = ({N^2}-1)+2.$

Now, let us see the statement for Fermat’s little theorem.

According to this theorem, if ‘a’ is not divisible by ‘p’ where ‘a’ and ‘p’ are integers, then the number ${a^{p - 1}} - 1$ is an integer multiple of ‘p’.

For example – if a= 2 and p = 7, then ${2^6} =64$ and $64 – 1 =63 = 7 \times 9$ is thus a multiple of 7.

So, using this theorem, we can say that $({N^2}-1)$ = ${N^{3 - 1}} - 1$ is divisible by 3.

But in$({N^2}-1)+2$, where we are adding 2, the number is now not divisible by 3.

That is ${N^2}+1$ is not divisible by 3. This contradicts our assumption.

**Hence, It is proved that no square number is of the form $3n – 1.$**

**Note:**This type of question is generally solved by contradiction method. In this method, the statement is proved by contradicting the assumption. In this question, you must remember Fermat's little theorem. This theorem can also be stated in other as follow:

If ‘p’ is a prime number then for any integer ‘a’, the number ${a^p}- a$ is an integer multiple of ‘p’.

For example- if $a =2$ and $p =7$ then ${2^7}= 128$, and $128 -2 = 126$, which is an integer multiple of 7.

Recently Updated Pages

Identify the type of clause underlined in the sentence class 8 english CBSE

Which statement describes the density of the inner class 8 social science CBSE

Babur considered which ruler of Gujarat as among the class 8 social science CBSE

Which island groups were affected by the Tsunami in class 8 social science CBSE

Which is the administrative system that works under class 8 social science CBSE

The year in which the state was named as Karnataka class 8 social science CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

List some examples of Rabi and Kharif crops class 8 biology CBSE

Which are the Top 10 Largest Countries of the World?

The provincial president of the constituent assembly class 11 social science CBSE

Write the 6 fundamental rights of India and explain in detail