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

Question

If x and y are positive integers, and x > y, then which of the following could be the value of x^2 - y^2 ?

I. 1
II. 10
III. 13

(A) I only
(B) II only
(C) III only
(D) II and III only
(E) I, II, and III

Jagmeister · Answer

My Answer: (C) 3 only

My Approach:

Given that x and y are positive integers and that x>y

if we start listing out the first few squares of positive numbers starting from 1

1,4,8,16,25,36,49..... so on.

The difference between consecutive squares is 3,4,8,9,11,13... so on

As we can observe, as we go higher in the series the difference between consecutive squares only increases.

Out of the given options we can see that options I and II are not possible with any combinations, but option III, i.e. 13 is possible with x=7 and y=6

Therefore   only option III is TRUE

kungfury42 · Answer

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.

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.

Posted from my mobile device

Riyaagoell · Answer

i dont understand, why 2 cant be correct

egmat · Answer

Riyaagoell

Looking at your question, I understand you're confused about why certain values can't work. Here's how I'd explain it:

Key Insight:   x^2 - y^2 = (x+y)(x-y)

Since  x  and  y  are positive integers with  x > y :
-  x + y  is a positive integer
-  x - y  is a positive integer (since  x > y )
- Therefore,  x^2 - y^2  must equal the product of two positive integers

Why option II (value 10) doesn't work: 
 (x+y)(x-y) = 10 
We can factor  10 = 1 	imes 10  or  10 = 2 	imes 5

Case 1:  x - y = 1, x + y = 10 
Solving:  2x = 11 , so  x = 5.5  ← Not an integer! ✗

Case 2:  x - y = 2, x + y = 5 
Solving:  2x = 7 , so  x = 3.5  ← Not an integer! ✗

What went wrong in your thinking: 
You likely didn't check whether the factorizations actually yield integer values for  x  and  y . Not every factorization of a number leads to valid integer solutions.

Here is a strategic framework for factorization test that you can apply:  
When you see " x^2 - y^2  with integer constraints":
1. Factor as  (x+y)(x-y) 
2. List all positive factor pairs
3. Check if  x = \frac{sum + difference}{2}  gives an integer
4. Only factor pairs where both sum and difference are the same parity (both odd or both even) will work

I hope this helps you. If you'd like, you can practice similar questions  here  (you'll find OG questions) - just select Number Properties under Quant and choose Easy level questions to build confidence with factorization patterns.

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

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