## Quant Question Of The Day: 28

[latexpage]

# Numbers

How many ordered integral pairs (x,y) satisfy the equation 3x^{2} – y^{2} = 1747 ?

A. 0

B. 1

C. 2

D. More than 2 pairs

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[latexpage]

How many ordered integral pairs (x,y) satisfy the equation 3x^{2} – y^{2} = 1747 ?

A. 0

B. 1

C. 2

D. More than 2 pairs

We know that every perfect square can be written as 3k or 3k+1 form.

If y^{2} is in the form of 3k then 3x^{2} – y^{2} will also be a multiple of 3.

If y^{2} is in the form of 3k+1 then 3x^{2} – y^{2} will be 2 more than a multiple of 3.

Now we can see 1747 gives 1 as a remainder when divided by 3,

Hence, 3x^{2} – y^{2} = 1747 will have no integer solution

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A. 0

Let’s look at this as 3x^2 = 1747 + y^2

Now, 1747 is of form 3k+1. Therefore, y^2 is of form 3k+2 for x^2 to be integral.

We know, square of any natural no. is of form 3k or 3k+1. Thus, no such solution possible.

No such pairs are possible.

Also ,is there any shortcut to ascertain which number is prime and which is not.Because my initial time got wasted in finding whether it is a prime number or not.

no such pair is possible as 1747 is a prime number (6K-1)

therefore answer A

Only one option is there i.e first find the square root then check with the prime no.below that square root ..if divided then it’s not else it is.

Only one option is there i.e first find the square root then check with the prime no.below that square root ..if divided then it’s not else it is……..and in this no such pair is possible

You find the square of the number closest to the given one; in this case 42 square = 1764.

Then you divide the given number i.e. 1747 with all the prime numbers below 42. If it is not divisible by any of these prime numbers, then the given number is also prime.

I use this method. If there is a shorter method than this then kindly share the same sir.

Sir please post the solution of this problem.

No pair possible