Number Operations and Properties

Applying the Pythagorean triples rule as $a^{2}+b^{2}= c^{2}$
Option A: $5^{2}+12^{2}= 13^{2}$
25 + 144 = 169
169 = 169 is a Pythagorean triples.

For any natural numbers m > 1, $2m, m^{2} - 1$, $m^{2} + 1$ form a Pythagorean triplet.
If we take $m^2 + 1 = 22$, then $m^2 = 21$
The value of m will not be an integer.
If we take $m^2 - 1 = 22$, then $m^2 = 23$
Again the value of m will not be an integer.
Let $2m = 22$
$m = \cfrac{22}{2}$
$m = 11$
$2m = 2 \times 11 = 22$
$m^{2} - 1$ = $11^{2} - 1$
$= 121 - 1 = 120$
$m^{2} + 1$ = $11^{2} + 1$
$= 121 + 1 = 122$
Therefore, the Pythagorean triplets are $ 22, 120, 122.$

$14, 12, 13$ is not a Pythagorean triplet.
On squaring, we get
$12^2 + 13^2\ and\ 14^2$
$144 + 169 \ and\ 196$
$313 \neq 196$

For any natural numbers m > 1, $2m, m^{2} - 1$, $m^{2} + 1$ forms a Pythagorean triplet.
If we take $m^2 + 1 = 34$, then $m^2 = 33$
The value of m will not be an integer.
If we take $m^2 - 1 = 34$, then $m^2 = 35$
Again the value of m will not be an integer.
Let $2m = 34$
$m = \dfrac{34}{2}$
$m = 17$
$2m = 2 \times 17 = 34$
$m^{2} - 1$ = $17^{2} - 1$
$= 289 - 1 = 288$
$m^{2} + 1$ = $17^{2} + 1$
$=289 + 1 = 290$
Therefore, the Pythagorean triplets are $34, 288, 290.$

$15, 114, 115$ is not a Pythagorean triplet.
On squaring, we get
$15^2 + 114^2 \ and \ 15^2$
$225 + 12996 \ and \  13225$
$13221 \neq 13225$

$\Rightarrow$  The given number is $31520332$.

(A) Place value of 5 in 
$91,25,678=5000$
Face value of 5 in 
$91,25,678=5$
$\therefore :difference=5000-5$
$=4995$ 

Face value of a digit in any place will be the digit itself.

Let the required number be $n$
Then, $3n+9=45$
$\Rightarrow$  $3n=45-9$
$\Rightarrow$  $3n=36$
$\Rightarrow$  $n=\cfrac{36}{3}=12$
Hence the required number is $12$

Place value of $5$ in
$91,25,678 = 5000$
Face value of $5$ in
$91,25,678 = 5$
$\displaystyle \therefore $ difference  $= 5000 - 5
= 4995$

$76,987$ reads as Seventy six thousand nine hundred eighty seven.

$88,789$ reads as Eighty eight thousand seven hundred eighty nine