Question

what is the value of x

Answer 1

Heya User,

 *Number theory*

--> We're willing to find out the integral soln.s for x, such that :->
-->   √(x² - 36) is an integer ..

Let √( x² - 36 ) = k
=>  ( x² - 36 ) = k²
=> x² - k² = 36
=> ( x + k )( x - k ) = 36

Noting the factors for 36:->
---> 36 = 1 * 36 ; 2 * 18 ; 3 * 12 ; 4 * ......

--> Eh! Note that :-> if ( x + k ) = odd , where ( x - k ) is also odd...

--> And, perhaps, 36 has no two odd divisors that altogether completely divide the number =_=

--> Hence, we note the even divisors :-> 
--> 36 = 2 * 18 ; 6 * 6 ; 
--> Comparing =_= ----->
----> [ x + k ] = 2 ; [ x - k ] = 18
----> [ x + k ] = 18 ; [ x - k ] = 2
----> [ x + k ] = 6 ; [ x - k ] = 6   

Solving the above eqn.s --> x = 10 || k = 8 ; or x = 6 ; k = 0

Hence, two possible soln.s are ---> ( x , k ) = ( 10 , 8 ) , ( 6 , 0 )