Question

How many non-square numbers lie between the following pairs of numbers?
(1) 7^2 and 8^2
(ii) 10^2 and 11^2
() 40^2 and 41^2
(iv) 80^2 and 81^2
(v) 101^2 and 102^2
(vi) 205^2 and 206^2


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Answer 1

Topic

• Algebra- Identity

An identity is an equation that is always true regardless of the values of the variables. We use it to factorize or rationalize the denominator.

• Algebra- Factorization

The method of showing a polynomial using the product of factors.

Solution

① Finding the pattern.

The answer is simple.

(1) 7+8-1=14

(2) 10+11-1=20

(3) 40+41-1=80

(4) 80+81-1=160

(5) 101+102-1=202

(6) 205+206-1=410

② What is the reason?

The reason behind this is identity. Let's take two integers $a$ and $b$ as an example.

$\implies a^{2}-b^{2}=(a+b)(a-b)$

It has two factors, $a+b$ and $a-b$. However, we can find $a-b$ here.

If $a$ is greater than $b$, the base of two numbers differ by 1 so $a-b=1$.

③ Multiplying by one.

Since multiplying by 1 doesn't change the number, $a^{2}-b^{2}=a+b$.

Now we subtract 1 from the result.

$\implies \boxed{a+b-1}$

The number of integers between two numbers can be found. This is how the pattern works.

This is the end of the required answer.

Recommended Examples

Question: How many non-square integers are between $30^{2}$ and $32^{2}$?

Answer:-

The number of integers between the two is $32^{2}-30^{2}-1$.

There is a perfect square $31^{2}$.

So, the answer will be,

$32^{2}-30^{2}-2$

$=(32+30)(32-30)-2$

$=62\cdot 2-2$

$=\boxed{122}$

There are 122 non-square numbers between $30^{2}$ and $32^{2}$.