Looking for harder GMAT quant problems to prep for your pursuit of 700+? Stay tuned to this space, where over the next few weeks we’ll continue to show you ways in which the GMAT can take its emphases on number properties, divisibility, and “creative algebra” to newer, harder heights. Today we offer a tricky problem solving question; enter your solutions and explanations in the comments field and we’ll be back later with a detailed solution.
0 < x < y and x and y are consecutive integers. If the difference between x2 and y2 = 12,201 then what is the value of x?
(A) 6,100
(B) 6,101
(C) 12,200
(D) 12,201
(E) 24,402
Solution:
The key to this seemingly labor-intensive problem is recognizing that you can apply the Difference of Squares rule. As the given information notes that y^2 – x^2 = 12,201, that also means that, by Difference of Squares:
(y + x)(y – x) = 12,201. And since we know that they are consecutive integers, that means that y = x + 1.
So, in those terms, (2x + 1)(1) = 12201, and so 2x = 12,200. therefore x = 6100.
The correct answer is (A).
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The answer is A. y²-x²=12,201
y²=12201+x²
=12200+x²+1
y²=2*6100+x²+1
Which ever value of x you plug in, the corresponding value of y² has to be a perfect square.
(A)Putting x=6,100
y²=2*6100+6100²+1…..This is of the form a²+2ab+b²=(a+b)²
y²=(6100+1)²=(6101)²
y=6101 Hence (A)
y^2 -x ^2= 12201 ( Since y is greater than x it is y^2 first)
(y+x) (y-x) = 12201
y+x=12201 ( Since y and x are consecutive y-x=1)
y=x+1( Since consecutive numbers)
2x=12201
x=6100.
(x^2 – y^2) = (x+y)(x-y)
Looking at the answer choice, and (x-y), we can deduce that (x-y) should be 1.
Hence (x+y)(x-y) = 12201
(x+y)*1 = 12201
x+y = 12201. Looking at the answer, choices a and b comes in scope. Also given that x<y hence x=6100. Answer – option a.
I did it a bit different.
My answer is A – 6100.
y=x+1
so
(x+1)^2 – x^2 = 2x+1 = 12201 —–> x=6100
y = x+1
y² = (x+1)²
y² – x² = (x+1)² – x² = x² + 2x + 1 – x² = 12201
2x+1 = 12201
x = (12201-1)/2 = 6100
since x and y are consecutive, then y=x+1
also y^2-x^2=12201
therefore, (x+1)^2-x^2=12201
2x=12200
x=6100 ANS
it’s so simple. much calculation not required.
just look at the problem for a while. if you do so, you will notice that the difference between y^2 and x^2 is also square and we have to find out the value of x, given that x<y and they are consecutive numbers.
from this , we can easily eliminate E,D,C and as x<y, so option is A.