If x and y are positive integers, and x > y, then which of the followi

Little algebraic but absolutely fail-proof method.

1. Let's write down our basic identity: (x+y)(x-y) = x² - y² Now we know that x² - y² = N is equivalent to saying (x+y)(x-y) = N

2. Since I know that x and y are positive integers and x>y can I split my N into two factors and compare the larger factor with (x+y) and smaller factor with (x-y)? Yes.

3. After that I can simply solve 2 very simple equations in 2 variables and see if I get meaningful values for x and y?

4. Before trying out the above, think of another interesting property. If my equations in step 3 are (x+y)=Bigger factor of N and (x-y)=Smaller factor of N, can I add these two equations and say 2x=Sum of those two factors of N and since we know that 2 is an even integer (x being anything) the sum of the two factors of N must be even.

Therefore, we get an important conclusion from here that the two factors in which I must split N should either be both ODD or both EVEN.

5. Let's start evaluating the options one by one.

(i) N=1, the only way I can split this is 1*1, the sum of the two factors is 1+1=2 and so the value of x is 1. But what about y? Y becomes (1-1=0) in this case and that violates our given condition that x and y are both +ve integers. Eliminate (i)

(ii) N=10, the only ways I can split N is 1*10 or 2*5 but both of these are invalid because 1+10=11 and 2+5=7 and both 11 and 7 are odd numbers and we know that the sum of the factors must be even for x and y to take integer values. Eliminate (ii)

(iii) If you've been all confidently careful upto this point, a quick glance at the options would tell you that there's honestly no need to evaluate N=13 because there is no option which says that neither of these three N are possible. But we'll still proceed with this case to demonstrate this beautiful method.

N=13, the only way I can split this into 2 factors is 1*13, and that is a valid factorization because 1+13=14 and 14 is an even number which will give me integral values of x and y.

Now we know that 2x = 14, which means x = 7 and substituting this back in any of the two equations we get y = (13-7) = 6

Let's verify x=7 and y=6
7² = 49
6² = 36

7²-6² = 49-36 = 13
Yes x² - y² = 13 is a valid possibility

Hence, our answer is option C, (iii) only.

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