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Algebra problems
by abhijit konwar - Friday, 1 August 2014, 12:16 PM
  How many integral values of (x,y) satisfy the equation

4x^2 - 9y^2 = 2100

do clarify the method.
Re: Algebra problems
by TG Team - Friday, 1 August 2014, 02:14 PM
  Hi Abhijit smile

4x2 - 9y2 = 2100
i.e. (2x - 3y)(2x + 3y) = 2100

So we need to write 2100 as product of two integers, both of which are either ODD or both EVEN {think why?}

Rest part is simple calculation.

Answer is 24. smile

Kamal Lohia
Re: Algebra problems
by abhijit konwar - Saturday, 2 August 2014, 03:47 PM
  thanks for the revert..But actually the answer is- no integral solutions.
Re: Algebra problems
by TG Team - Monday, 4 August 2014, 11:58 AM
  Hi Abhijit smile

Thanks for the correction. Yes you are right that there are no integral solutions.

4x2 - 9y2 = 2100
i.e. (2x - 3y)(2x + 3y) = 2100

In this part I know that difference between two integers on the LHS is 6y i.e. even, so that means RHS must be able to be written as product of two even integers or two odd integers and that is possible as RHS is multiple of 4.

But what I missed was the other fact that difference between two integers on the LHS is 6y i.e. multiple of 3 too. So the RHS must be able to be written as product of two numbers both of which are multiple of 3 (i.e. RHS being multiple of 9, which is not the case) or both of which are not multiple of 3 (which is also not the case).

Thus there are NO integral solutions to given equation. smile

Kamal Lohia 
Re: Algebra problems
by SaiKiran Allenki - Thursday, 21 August 2014, 07:07 PM
  Hi,
4x^2 - 9y^2 = 2100

2100 => 2^2 * 5^2 * 7 * 3
Now,
(2x+3y) ( 2x-3y) =
2^2 * 5^2 * 7 * 3

if we add the factors on the LHS the result obtained is 4x, therefore the two 2's cannot be assigned to one bracket.
i.e it should be in the form of
(2*A) (2*B)
let us see this in other way, say you assigned two 2's to 1st bracket
(4*A)(this bracket should have an even no(which we do not have in this case)) because the sum of 2 brackets is 4x which is even.

now,
if we take the difference on the LHS the result obtained is 6y again an even and a multiple of 3. So the RHS must be written as product of 2 no's in the brackets which are multiples of 3, which is not possible in this case.


therefore no solution.
Hope this helps you.
Re: Algebra problems
by piyush S - Saturday, 4 October 2014, 04:22 AM
  Sir , for these type of questions is there any short way to find all solution set?(if it exist) not about this question but generally?