Question

Find the number of positive integers less than 101 that cannot be written as the difference of two squares of integers.

Answer 1

Let $c$ be a positive integer less than $101$ such that $c=a^2-b^2$
Now, $c=a^2-b^2\Rightarrow c=(a+b)(a-b)$

$\text{Case I :}a-b=1$
$\therefore c=a+b$
Then, for $c$ to be an integer $a+b$ must be an odd integer.
Hence, $c$ can be an odd integer.

$\text{Case II :}a-b=2$
$\therefore c=2(a+b)$
Then, for $c$ to be an integer $a+b$ must be an even integer.
Therefore, $c$ will be a multiple of $4$
Hence, $c$ can be a multiple of $4$.

Now, if $a-b=3,5,7,....$ any odd integer.
Then, for $c$ to be an integer $a+b$ must be an odd integer, which all have already been counted in $\text{Case I}$

Now, if $a-b=4,5,7,....$ any odd integer.
Then, $c$ will be a multiple of $4$, which all have already been counted in $\text{Case II}$

$\therefore$ No. of numbers which can be expressed as difference of squares
= (odd numbers) + (multiples of 4) = 50 + 25 = 75
$\therefore$ Required numbers which cannot be expressed as a difference of squares of two integers are = 100 - 75 = 25.


Alternate Solution :
Let $c\to$ be positive integer $\le 100$ such that $c=a^2-b^2$; $a,b\in Z$
$C-I$: Difference of $a$ and $b$ is 1.
$\therefore 1^2-0^2=1$
$2^2-1^2=3$
$3^2-2^2=5$
$4^2-3^2=7$
$5^2-4^2=9$
.
.
.
${50}^2-{49}^2=99$
All odd number upto 100 can be expressed as difference of two squares.

$C-II$: Difference of $a$ and $b$ is 2.
$2^2-0^2=4$
$3^2-1^2=8$
$4^2-2^2=12$
.
.
.
${26}^2-{24}^2=100$
All multiplies of '4' upto 100 can be expressed as difference of two squares.

$C-III$: Difference of $a,b$ is 3
This case will give odd numbers which are already counted

$C-IV$: Difference of $a,b$ is 4
This case will give multiples of 4 which are already counted
Similarly, the remaining cases will all give number already counted in $C-I$ and $C-II$
$\therefore$ No. of numbers which can be expressed as difference of squares
= (odd numbers) + (multiples of 4) = 50 + 25 = 75
$\therefore$ Required numbers which cannot be expressed as a difference of squares of two integers are = 100 - 75 = 25.